text: from __future__ import division
  
  
  from sympy import (acos, acosh, asinh, atan, cos, Derivative, diff, dsolve, Eq,
                     erf, exp, Function, I, Integral, LambertW, log, O, pi,
                     Rational, RootOf, S, simplify, sin, sqrt, Symbol, tan, asin,
                     Piecewise, symbols)
  x,y,z = symbols('x:z', real=True)
  from sympy.solvers.ode import (_undetermined_coefficients_match, checkodesol,
                                 classify_ode, constant_renumber, constantsimp,
                                 homogeneous_order, ode_order)
  from sympy.utilities.pytest import XFAIL, skip, raises, slow
  
  C1, C2, C3, C4, C5, C6, C7, C8, C9, C10 = symbols('C1:11')
  f = Function('f')
  g = Function('g')
  
  # Note: the tests below may fail (but still be correct) if ODE solver,
  # the integral engine, solve(), or even simplify() changes. Also, in
  # differently formatted solutions, the arbitrary constants might not be
  # equal.  Using specific hints in tests can help to avoid this.
  
  # Tests of order higher than 1 should run the solutions through
  # constant_renumber because it will normalize it (constant_renumber causes
  # dsolve() to return different results on different machines)
  
  def test_checkodesol():
      # For the most part, checkodesol is well tested in the tests below.
      # These tests only handle cases not checked below.
      raises(ValueError, 'checkodesol(f(x, y).diff(x), Eq(f(x, y), x))')
      raises(ValueError, 'checkodesol(f(x).diff(x), Eq(f(x, y), x), f(x, y))')
      assert checkodesol(f(x).diff(x), Eq(f(x, y), x)) == \
          (False, -f(x).diff(x) + f(x, y).diff(x) - 1)
      assert checkodesol(f(x).diff(x), Eq(f(x), x)) is not True
      assert checkodesol(f(x).diff(x), Eq(f(x), x)) == (False, 1)
      sol1 = Eq(f(x)**5 + 11*f(x) - 2*f(x) + x, 0)
      assert checkodesol(diff(sol1.lhs, x), sol1) == (True, 0)
      assert checkodesol(diff(sol1.lhs, x)*exp(f(x)), sol1) == (True, 0)
      assert checkodesol(diff(sol1.lhs, x, 2), sol1) == (True, 0)
      assert checkodesol(diff(sol1.lhs, x, 2)*exp(f(x)), sol1) == (True, 0)
      assert checkodesol(diff(sol1.lhs, x, 3), sol1) == (True, 0)
      assert checkodesol(diff(sol1.lhs, x, 3)*exp(f(x)), sol1) == (True, 0)
      assert checkodesol(diff(sol1.lhs, x, 3), Eq(f(x), x*log(x))) == \
          (False, -9 + 60*x**4*log(x)**2 + 240*x**4*log(x)**3 +
          235*x**4*log(x)**4 + 60*x**4*log(x)**5)
      assert checkodesol(diff(exp(f(x)) + x, x)*x, Eq(exp(f(x)) + x)) == \
          (True, 0)
      assert checkodesol(diff(exp(f(x)) + x, x)*x, Eq(exp(f(x)) + x), \
          solve_for_func=False) == (True, 0)
      assert checkodesol(f(x).diff(x, 2), [Eq(f(x), C1 + C2*x), \
          Eq(f(x), C2 + C1*x), Eq(f(x), C1*x + C2*x**2)]) == \
              [(True, 0), (True, 0), (False, 2*C2)]
      assert checkodesol(f(x).diff(x, 2), set([Eq(f(x), C1 + C2*x), \
          Eq(f(x), C2 + C1*x), Eq(f(x), C1*x + C2*x**2)])) == \
              set([(True, 0), (True, 0), (False, 2*C2)])
      assert checkodesol(f(x).diff(x) - 1/f(x)/2, Eq(f(x)**2, x)) == \
          [(True, 0), (True, 0)]
      assert checkodesol(f(x).diff(x) - f(x), Eq(C1*exp(x), f(x))) == (True, 0)
      # Based on test_1st_homogeneous_coeff_ode2_eq3sol.  Make sure that
      # checkodesol tries back substituting f(x) when it can.
      eq3 = x*exp(f(x)/x) + f(x) - x*f(x).diff(x)
      sol3 = Eq(f(x), log(log(C1/x)**(-x)))
      assert not checkodesol(eq3, sol3)[1].has(f(x))
  
  def test_dsolve_options():
      eq = x*f(x).diff(x) + f(x)
      a = dsolve(eq, hint='all')
      b = dsolve(eq, hint='all', simplify=False)
      c = dsolve(eq, hint='all_Integral')
      keys = ['1st_exact', '1st_exact_Integral', '1st_homogeneous_coeff_best', \
          '1st_homogeneous_coeff_subs_dep_div_indep', \
          '1st_homogeneous_coeff_subs_dep_div_indep_Integral', \
          '1st_homogeneous_coeff_subs_indep_div_dep', \
          '1st_homogeneous_coeff_subs_indep_div_dep_Integral', '1st_linear', \
          '1st_linear_Integral', 'best', 'best_hint', 'default', 'order', \
          'separable', 'separable_Integral']
      Integral_keys = ['1st_exact_Integral',
      '1st_homogeneous_coeff_subs_dep_div_indep_Integral',
      '1st_homogeneous_coeff_subs_indep_div_dep_Integral', '1st_linear_Integral',
      'best', 'best_hint', 'default', 'order', 'separable_Integral']
      assert sorted(a.keys()) == keys
      assert a['order'] == ode_order(eq, f(x))
      assert a['best'] == Eq(f(x), C1/x)
      assert dsolve(eq, hint='best') == Eq(f(x), C1/x)
      assert a['default'] == 'separable'
      assert a['best_hint'] == 'separable'
      assert not a['1st_exact'].has(Integral)
      assert not a['separable'].has(Integral)
      assert not a['1st_homogeneous_coeff_best'].has(Integral)
      assert not a['1st_homogeneous_coeff_subs_dep_div_indep'].has(Integral)
      assert not a['1st_homogeneous_coeff_subs_indep_div_dep'].has(Integral)
      assert not a['1st_linear'].has(Integral)
      assert a['1st_linear_Integral'].has(Integral)
      assert a['1st_exact_Integral'].has(Integral)
      assert a['1st_homogeneous_coeff_subs_dep_div_indep_Integral'].has(Integral)
      assert a['1st_homogeneous_coeff_subs_indep_div_dep_Integral'].has(Integral)
      assert a['separable_Integral'].has(Integral)
      assert sorted(b.keys()) == keys
      assert b['order'] == ode_order(eq, f(x))
      assert b['best'] == Eq(f(x), C1/x)
      assert dsolve(eq, hint='best', simplify=False) == Eq(f(x), C1/x)
      assert b['default'] == 'separable'
      assert b['best_hint'] == '1st_linear'
      assert a['separable'] != b['separable']
      assert a['1st_homogeneous_coeff_subs_dep_div_indep'] != \
          b['1st_homogeneous_coeff_subs_dep_div_indep']
      assert a['1st_homogeneous_coeff_subs_indep_div_dep'] != \
          b['1st_homogeneous_coeff_subs_indep_div_dep']
      assert not b['1st_exact'].has(Integral)
      assert not b['separable'].has(Integral)
      assert not b['1st_homogeneous_coeff_best'].has(Integral)
      assert not b['1st_homogeneous_coeff_subs_dep_div_indep'].has(Integral)
      assert not b['1st_homogeneous_coeff_subs_indep_div_dep'].has(Integral)
      assert not b['1st_linear'].has(Integral)
      assert b['1st_linear_Integral'].has(Integral)
      assert b['1st_exact_Integral'].has(Integral)
      assert b['1st_homogeneous_coeff_subs_dep_div_indep_Integral'].has(Integral)
      assert b['1st_homogeneous_coeff_subs_indep_div_dep_Integral'].has(Integral)
      assert b['separable_Integral'].has(Integral)
      assert sorted(c.keys()) == Integral_keys
      raises(ValueError, "dsolve(eq, hint='notarealhint')")
      raises(ValueError, "dsolve(eq, hint='Liouville')")
      assert dsolve(f(x).diff(x) - 1/f(x)**2, hint='all')['best'] == \
          dsolve(f(x).diff(x) - 1/f(x)**2, hint='best')
      assert dsolve(f(x) + f(x).diff(x) + sin(x).diff(x) + 1, f(x),
                    hint="1st_linear_Integral") == \
          Eq(f(x), (C1 + Integral((-sin(x).diff(x) - 1)*
                  exp(Integral(1, x)), x))*exp(-Integral(1, x)))
  def test_classify_ode():
      assert classify_ode(f(x).diff(x, 2), f(x)) == \
          ('nth_linear_constant_coeff_homogeneous', 'Liouville',
              'Liouville_Integral')
      assert classify_ode(f(x), f(x)) == ()
      assert classify_ode(Eq(f(x).diff(x), 0), f(x)) == ('separable',
          '1st_linear', '1st_homogeneous_coeff_best',
          '1st_homogeneous_coeff_subs_indep_div_dep',
          '1st_homogeneous_coeff_subs_dep_div_indep',
          'nth_linear_constant_coeff_homogeneous',
          'separable_Integral',
          '1st_linear_Integral',
          '1st_homogeneous_coeff_subs_indep_div_dep_Integral',
          '1st_homogeneous_coeff_subs_dep_div_indep_Integral')
      assert classify_ode(f(x).diff(x)**2, f(x)) == ()
      # 1650: f(x) should be cleared from highest derivative before classifying
      a = classify_ode(Eq(f(x).diff(x) + f(x), x), f(x))
      b = classify_ode(f(x).diff(x)*f(x) + f(x)*f(x) - x*f(x), f(x))
      c = classify_ode(f(x).diff(x)/f(x) + f(x)/f(x) - x/f(x), f(x))
      assert a == b == c != ()
      assert classify_ode(
          2*x*f(x)*f(x).diff(x) + (1 + x)*f(x)**2 - exp(x), f(x)
          ) == ('Bernoulli', 'Bernoulli_Integral')
      assert 'Riccati_special_minus2' in\
          classify_ode(2*f(x).diff(x) + f(x)**2 - f(x)/x + 3*x**(-2), f(x))
      raises(ValueError, "classify_ode(x + f(x, y).diff(x).diff(y), f(x, y))")
      # 2077
      k = Symbol('k')
      assert classify_ode(f(x).diff(x)/(k*f(x) + k*x*f(x)) +
                   2*f(x)/(k*f(x) + k*x*f(x)) +
                   x*f(x).diff(x)/(k*f(x) + k*x*f(x))
                   + z, f(x)) == ('separable',
                                  '1st_exact',
                                  'separable_Integral',
                                  '1st_exact_Integral')
      # preprocessing
      ans = ('separable', '1st_exact', '1st_linear', 'Bernoulli',
          '1st_homogeneous_coeff_best',
          '1st_homogeneous_coeff_subs_indep_div_dep',
          '1st_homogeneous_coeff_subs_dep_div_indep',
          'nth_linear_constant_coeff_undetermined_coefficients',
          'nth_linear_constant_coeff_variation_of_parameters',
          'separable_Integral', '1st_exact_Integral',
          '1st_linear_Integral',
          'Bernoulli_Integral',
          '1st_homogeneous_coeff_subs_indep_div_dep_Integral',
         '1st_homogeneous_coeff_subs_dep_div_indep_Integral',
         'nth_linear_constant_coeff_variation_of_parameters_Integral')
      #     w/o f(x) given
      assert classify_ode(diff(f(x) + x, x) + diff(f(x), x)) == ans
      #     w/ f(x) and prep=True
      assert classify_ode(diff(f(x) + x, x) + diff(f(x), x), f(x),
                          prep=True) == ans
  
  def test_ode_order():
      f = Function('f')
      g = Function('g')
      x = Symbol('x')
      assert ode_order(3*x*exp(f(x)), f(x)) == 0
      assert ode_order(x*diff(f(x),x)+3*x*f(x)-sin(x)/x, f(x)) == 1
      assert ode_order(x**2*f(x).diff(x,x)+x*diff(f(x),x)-f(x),f(x)) == 2
      assert ode_order(diff(x*exp(f(x)),x,x), f(x)) == 2
      assert ode_order(diff(x*diff(x*exp(f(x)), x,x), x), f(x)) == 3
      assert ode_order(diff(f(x), x, x), g(x)) == 0
      assert ode_order(diff(f(x), x, x)*diff(g(x), x), f(x)) == 2
      assert ode_order(diff(f(x), x, x)*diff(g(x), x), g(x)) == 1
      assert ode_order(diff(x*diff(x*exp(f(x)), x,x), x), g(x)) == 0
      # issue 2736: ode_order has to also work for unevaluated derivatives
      # (ie, without using doit()).
      assert ode_order(Derivative(x*f(x), x), f(x)) == 1
      assert ode_order(x*sin(Derivative(x*f(x)**2, x, x)), f(x)) == 2
      assert ode_order(Derivative(x*Derivative(x*exp(f(x)), x,x), x), g(x)) == 0
      assert ode_order(Derivative(f(x), x, x), g(x)) == 0
      assert ode_order(Derivative(x*exp(f(x)),x,x), f(x)) == 2
      assert ode_order(Derivative(f(x), x, x)*Derivative(g(x), x), g(x)) == 1
      assert ode_order(Derivative(x*Derivative(f(x), x, x), x), f(x)) == 3
      assert ode_order(
          x*sin(Derivative(x*Derivative(f(x), x)**2, x, x)), f(x)) == 3
  
  
  # In all tests below, checkodesol has the order option set to prevent
  # superfluous calls to ode_order(), and the solve_for_func flag set to False
  # because dsolve() already tries to solve for the function, unless the
  # simplify=False option is set.
  def test_old_ode_tests():
      # These are simple tests from the old ode module
      eq1 = Eq(f(x).diff(x), 0)
      eq2 = Eq(3*f(x).diff(x) - 5, 0)
      eq3 = Eq(3*f(x).diff(x), 5)
      eq4 = Eq(9*f(x).diff(x, x) + f(x), 0)
      eq5 = Eq(9*f(x).diff(x, x), f(x))
      # Type: a(x)f'(x)+b(x)*f(x)+c(x)=0
      eq6 = Eq(x**2*f(x).diff(x) + 3*x*f(x) - sin(x)/x, 0)
      eq7 = Eq(f(x).diff(x,x) - 3*diff(f(x),x) + 2*f(x), 0)
      # Type: 2nd order, constant coefficients (two real different roots)
      eq8 = Eq(f(x).diff(x,x) - 4*diff(f(x),x) + 4*f(x), 0)
      # Type: 2nd order, constant coefficients (two real equal roots)
      eq9 = Eq(f(x).diff(x,x)+2*diff(f(x),x)+3*f(x), 0)
      # Type: 2nd order, constant coefficients (two complex roots)
      eq10 = Eq(3*f(x).diff(x) -1,0)
      eq11 = Eq(x*f(x).diff(x) -1,0)
      sol1 = Eq(f(x), C1)
      sol2 = Eq(f(x), C1+5*x/3)
      sol3 = Eq(f(x), C1+5*x/3)
      sol4 = Eq(f(x), C1*sin(x/3) + C2*cos(x/3))
      sol5 = Eq(f(x), C1*exp(-x/3) + C2*exp(x/3))
      sol6 = Eq(f(x), (C1-cos(x))/x**3)
      sol7 = Eq(f(x), C1*exp(x) + C2*exp(2*x))
      sol8 = Eq(f(x), (C1 + C2*x)*exp(2*x))
      sol9 = Eq(f(x), (C1*sin(x*sqrt(2)) + C2*cos(x*sqrt(2)))*exp(-x))
      sol10 = Eq(f(x), C1 + x/3)
      sol11 = Eq(f(x), C1 + log(x))
      assert dsolve(eq1) == sol1
      assert dsolve(eq1.lhs) == sol1
      assert dsolve(eq2) == sol2
      assert dsolve(eq3) == sol3
      assert dsolve(eq4) == sol4
      assert dsolve(eq5) == sol5
      assert dsolve(eq6) == sol6
      assert dsolve(eq7) == sol7
      assert dsolve(eq8) == sol8
      assert dsolve(eq9) == sol9
      assert dsolve(eq10) == sol10
      assert dsolve(eq11) == sol11
      assert checkodesol(eq1, sol1, order=1, solve_for_func=False)[0]
      assert checkodesol(eq2, sol2, order=1, solve_for_func=False)[0]
      assert checkodesol(eq3, sol3, order=1, solve_for_func=False)[0]
      assert checkodesol(eq4, sol4, order=2, solve_for_func=False)[0]
      assert checkodesol(eq5, sol5, order=2, solve_for_func=False)[0]
      assert checkodesol(eq6, sol6, order=1, solve_for_func=False)[0]
      assert checkodesol(eq7, sol7, order=2, solve_for_func=False)[0]
      assert checkodesol(eq8, sol8, order=2, solve_for_func=False)[0]
      assert checkodesol(eq9, sol9, order=2, solve_for_func=False)[0]
      assert checkodesol(eq10, sol10, order=1, solve_for_func=False)[0]
      assert checkodesol(eq11, sol11, order=1, solve_for_func=False)[0]
  
  def test_1st_linear():
      # Type: first order linear form f'(x)+p(x)f(x)=q(x)
      eq = Eq(f(x).diff(x) + x*f(x), x**2)
      sol = Eq(f(x),exp(-x**2/2)*(sqrt(2)*sqrt(pi)*I*erf(I*x/sqrt(2))/2 \
      + x*exp(x**2/2) + C1))
      assert dsolve(eq, hint='1st_linear') == sol
      assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
  
  
  def test_Bernoulli():
      # Type: Bernoulli, f'(x) + p(x)*f(x) == q(x)*f(x)**n
      eq = Eq(x*f(x).diff(x) + f(x) - f(x)**2,0)
      sol = dsolve(eq,f(x), hint='Bernoulli')
      assert sol == Eq(f(x),1/(x*(C1 + 1/x)))
      assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
  
  def test_Riccati_special_minus2():
      # Type: Riccati special alpha = -2, a*dy/dx + b*y**2 + c*y/x +d/x**2
      eq = 2*f(x).diff(x) + f(x)**2 - f(x)/x + 3*x**(-2)
      sol = dsolve(eq,f(x), hint='Riccati_special_minus2')
      assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
  
  def test_1st_exact1():
      # Type: Exact differential equation, p(x,f) + q(x,f)*f' == 0,
      # where dp/df == dq/dx
      eq1 = sin(x)*cos(f(x)) + cos(x)*sin(f(x))*f(x).diff(x)
      eq2 = (2*x*f(x) + 1)/f(x) + (f(x) - x)/f(x)**2*f(x).diff(x)
      eq3 = 2*x + f(x)*cos(x) + (2*f(x) + sin(x) - sin(f(x)))*f(x).diff(x)
      eq4 = cos(f(x)) - (x*sin(f(x)) - f(x)**2)*f(x).diff(x)
      eq5 = 2*x*f(x) + (x**2 + f(x)**2)*f(x).diff(x)
      sol1 = Eq(f(x),acos((C1)/cos(x)))
      sol2 = Eq(log(f(x))+x/f(x)+x**2,C1)
      sol3 = Eq(f(x)*sin(x)+cos(f(x))+x**2+f(x)**2,C1)
      sol4 = Eq(x*cos(f(x))+f(x)**3/3,C1)
      sol5 = Eq(x**2*f(x) + f(x)**3/3, C1)
      assert dsolve(eq1,f(x), hint='1st_exact') == sol1
      assert dsolve(eq2,f(x), hint='1st_exact') == sol2
      assert dsolve(eq3,f(x), hint='1st_exact') == sol3
      assert dsolve(eq4, hint='1st_exact') == sol4
      assert dsolve(eq5, hint='1st_exact', simplify=False) == sol5
      assert checkodesol(eq1, sol1, order=1, solve_for_func=False)[0]
      assert checkodesol(eq2, sol2, order=1, solve_for_func=False)[0]
      assert checkodesol(eq3, sol3, order=1, solve_for_func=False)[0]
      assert checkodesol(eq4, sol4, order=1, solve_for_func=False)[0]
      assert checkodesol(eq5, sol5, order=1, solve_for_func=False)[0]
  
  @slow
  @XFAIL
  def test_1st_exact2():
      """
      This is an exact equation that fails under the exact engine. It is caught
      by first order homogeneous albeit with a much contorted solution.  The
      exact engine fails because of a poorly simplified integral of q(0,y)dy,
      where q is the function multiplying f'.  The solutions should be
      Eq(sqrt(x**2+f(x)**2)**3+y**3, C1).  The equation below is
      equivalent, but it is so complex that checkodesol fails, and takes a long
      time to do so.
      """
      eq = (x*sqrt(x**2 + f(x)**2) - (x**2*f(x)/(f(x) -
            sqrt(x**2 + f(x)**2)))*f(x).diff(x))
      sol = dsolve(eq)
      assert sol == Eq(log(x),
          C1 - 9*sqrt(1 + f(x)**2/x**2)*asinh(f(x)/x)/(-27*f(x)/x +
          27*sqrt(1 + f(x)**2/x**2)) - 9*sqrt(1 + f(x)**2/x**2)*
          log(1 - sqrt(1 + f(x)**2/x**2)*f(x)/x + 2*f(x)**2/x**2)/
          (-27*f(x)/x + 27*sqrt(1 + f(x)**2/x**2)) +
          9*asinh(f(x)/x)*f(x)/(x*(-27*f(x)/x + 27*sqrt(1 + f(x)**2/x**2))) +
          9*f(x)*log(1 - sqrt(1 + f(x)**2/x**2)*f(x)/x + 2*f(x)**2/x**2)/
          (x*(-27*f(x)/x + 27*sqrt(1 + f(x)**2/x**2))))
      assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
  
  def test_separable1():
      # test_separable1-5 are from Ordinary Differential Equations, Tenenbaum and
      # Pollard, pg. 55
      eq1 = f(x).diff(x) - f(x)
      eq2 = x*f(x).diff(x) - f(x)
      eq3 = f(x).diff(x) + sin(x)
      eq4 = f(x)**2 + 1 - (x**2 + 1)*f(x).diff(x)
      eq5 = f(x).diff(x)/tan(x) - f(x) - 2
      sol1 = Eq(f(x), C1*exp(x))
      sol2 = Eq(f(x), C1*x)
      sol3 = Eq(f(x), C1 + cos(x))
      sol4 = Eq(atan(f(x)), C1 + atan(x))
      sol5 = Eq(f(x), -2 + C1*sqrt(1 + tan(x)**2))
      #sol5 = Eq(f(x), C1*(C2 + sqrt(1 + tan(x)**2)))
      #sol5 = Eq(-log(2 + f(x)), C1 - log(1 + tan(x)**2)/2)
      assert dsolve(eq1, hint='separable') == sol1
      assert dsolve(eq2, hint='separable') == sol2
      assert dsolve(eq3, hint='separable') == sol3
      assert dsolve(eq4, hint='separable', simplify=False) == sol4
      assert dsolve(eq5, hint='separable') == simplify(sol5)
      assert checkodesol(eq1, sol1, order=1, solve_for_func=False)[0]
      assert checkodesol(eq2, sol2, order=1, solve_for_func=False)[0]
      assert checkodesol(eq3, sol3, order=1, solve_for_func=False)[0]
      assert checkodesol(eq4, sol4, order=1, solve_for_func=False)[0]
      assert checkodesol(eq5, sol5, order=1, solve_for_func=False)[0]
  
  def test_separable2():
      a = Symbol('a')
      eq6 = f(x)*x**2*f(x).diff(x) - f(x)**3 - 2*x**2*f(x).diff(x)
      eq7 = f(x)**2 - 1 - (2*f(x) + x*f(x))*f(x).diff(x)
      eq8 = x*log(x)*f(x).diff(x) + sqrt(1 + f(x)**2)
      eq9 = exp(x + 1)*tan(f(x)) + cos(f(x))*f(x).diff(x)
      eq10 = (x*cos(f(x)) + x**2*sin(f(x))*f(x).diff(x) -
              a**2*sin(f(x))*f(x).diff(x))
      # solve() messes this one up a little bit, so lets test _Integral here
      # We have to test strings with _Integral because y is a dummy variable.
      sol6str = ("Integral((_y - 2)/_y**3, (_y, f(x))) "
                  "== C1 + Integral(x**(-2), x)")
      sol7 = Eq(-log(-1 + f(x)**2)/2, C1 - log(2 + x))
      sol8 = Eq(asinh(f(x)), C1 - log(log(x)))
      # integrate cannot handle the integral on the lhs (cos/tan)
      sol9str = ("Integral(cos(_y)/tan(_y), (_y, f(x)))"
                 " == C1 + Integral(-E*exp(x), x)")
      sol10 = Eq(-log(-1 + sin(f(x))**2)/2, C1 - log(x**2 - a**2)/2)
      assert str(dsolve(eq6, hint='separable_Integral')) == sol6str
      assert dsolve(eq7, hint='separable', simplify=False) == sol7
      assert dsolve(eq8, hint='separable', simplify=False) == sol8
      assert str(dsolve(eq9, hint='separable_Integral')) == sol9str
      assert dsolve(eq10, hint='separable', simplify=False) == sol10
      assert checkodesol(eq7, sol7, order=1, solve_for_func=False)[0]
      assert checkodesol(eq8, sol8, order=1, solve_for_func=False)[0]
      assert checkodesol(eq10, sol10, order=1, solve_for_func=False)[0]
  
  def test_separable3():
      eq11 = f(x).diff(x) - f(x)*tan(x)
      eq12 = (x - 1)*cos(f(x))*f(x).diff(x) - 2*x*sin(f(x))
      eq13 = f(x).diff(x) - f(x)*log(f(x))/tan(x)
      sol11 = Eq(f(x), C1*sqrt(1 + tan(x)**2))
      sol12 = Eq(log(-1 + cos(f(x))**2)/2, C1 + 2*x + 2*log(x - 1))
      sol13 = Eq(log(log(f(x))), C1 - log(1 + tan(x)**2)/2 + log(tan(x)))
      assert dsolve(eq11, hint='separable') == simplify(sol11)
      assert dsolve(eq12, hint='separable', simplify=False) == sol12
      assert dsolve(eq13, hint='separable', simplify=False) == sol13
      assert checkodesol(eq11, sol11, order=1, solve_for_func=False)[0]
      assert checkodesol(eq13, sol13, order=1, solve_for_func=False)[0]
  
  def test_separable4():
      # This has a slow integral (1/((1 + y**2)*atan(y))), so we isolate it.
      eq14 = x*f(x).diff(x) + (1 + f(x)**2)*atan(f(x))
      sol14 = Eq(log(atan(f(x))), C1 - log(x))
      assert dsolve(eq14, hint='separable', simplify=False) == sol14
      assert checkodesol(eq14, sol14, order=1, solve_for_func=False)[0]
  
  def test_separable5():
      eq15 = f(x).diff(x) + x*(f(x) + 1)
      eq16 = exp(f(x)**2)*(x**2 + 2*x + 1) + (x*f(x) + f(x))*f(x).diff(x)
      eq17 = f(x).diff(x) + f(x)
      eq18 = sin(x)*cos(2*f(x)) + cos(x)*sin(2*f(x))*f(x).diff(x)
      eq19 = (1 - x)*f(x).diff(x) - x*(f(x) + 1)
      eq20 = f(x)*diff(f(x), x) + x - 3*x*f(x)**2
      eq21 = f(x).diff(x) - exp(x + f(x))
      sol15 = Eq(f(x), -1 + C1*exp(-x**2/2))
      sol16 = Eq(-exp(-f(x)**2)/2, C1 - x - x**2/2)
      sol17 = Eq(f(x), C1*exp(-x))
      sol18 = Eq(-log(-1 + sin(2*f(x))**2)/4, C1 + log(-1 + sin(x)**2)/2)
      sol19a = Eq(f(x), -(1 - x - C1*exp(-x))/(1 - x))
      sol19b = Eq(f(x), -1 + C1*exp(-x)/(-1 + x))
      sol19c = Eq(f(x), -1/(1 - x) + x/(1 - x) + C1*exp(-x)/(1 - x))
      sol19d = Eq(f(x), (C1*(1 - x) + x*(-x*exp(x) + exp(x))- exp(x) + x*exp(x))/
                          ((1 - x)*(-x*exp(x) + exp(x))))
      sol19e = Eq(f(x), (C1*(1 - x) - x*(-x*exp(x) + exp(x)) -
                              x*exp(x) + exp(x))/((1 - x)*(-exp(x) + x*exp(x))))
      sol19f = Eq(f(x), (C1 + (x - 1)*exp(x))*exp(-x)/(-x + 1))
      sol20 = Eq(log(-1 + 3*f(x)**2)/6, C1 + x**2/2)
      sol21 = Eq(-exp(-f(x)), C1 + exp(x))
      assert dsolve(eq15, hint='separable') == sol15
      assert dsolve(eq16, hint='separable', simplify=False) == sol16
      assert dsolve(eq17, hint='separable') == sol17
      assert dsolve(eq18, hint='separable', simplify=False) == sol18
      assert dsolve(eq19, hint='separable') in [sol19f, sol19a, sol19b, sol19c,
                                                sol19d, sol19e]
      assert dsolve(eq20, hint='separable', simplify=False) == sol20
      assert dsolve(eq21, hint='separable', simplify=False) == sol21
      assert checkodesol(eq15, sol15, order=1, solve_for_func=False)[0]
      assert checkodesol(eq16, sol16, order=1, solve_for_func=False)[0]
      assert checkodesol(eq17, sol17, order=1, solve_for_func=False)[0]
      assert checkodesol(eq18, sol18, order=1, solve_for_func=False)[0]
      assert checkodesol(eq19, sol19a, order=1, solve_for_func=False)[0]
      assert checkodesol(eq20, sol20, order=1, solve_for_func=False)[0]
      assert checkodesol(eq21, sol21, order=1, solve_for_func=False)[0]
  
  def test_separable_1_5_checkodesol():
      eq12 = (x - 1)*cos(f(x))*f(x).diff(x) - 2*x*sin(f(x))
      sol12 = Eq(-log(1 - cos(f(x))**2)/2, C1 - 2*x - 2*log(1 - x))
      assert checkodesol(eq12, sol12, order=1, solve_for_func=False)[0]
  
  def test_homogeneous_order():
      assert homogeneous_order(exp(y/x) + tan(y/x), x, y) == 0
      assert homogeneous_order(x**2 + sin(x)*cos(y), x, y) == None
      assert homogeneous_order(x - y - x*sin(y/x), x, y) == 1
      assert homogeneous_order((x*y + sqrt(x**4+y**4) + x**2*(log(x) - log(y)))/\
          (pi*x**Rational(2,3)*sqrt(y)**3), x, y) == Rational(-1,6)
      assert homogeneous_order(y/x*cos(y/x) - x/y*sin(y/x) + cos(y/x), x, y) == 0
      assert homogeneous_order(f(x), x, f(x)) == 1
      assert homogeneous_order(f(x)**2, x, f(x)) == 2
      assert homogeneous_order(x*y*z, x, y) == 2
      assert homogeneous_order(x*y*z, x, y, z) == 3
      assert homogeneous_order(x**2*f(x)/sqrt(x**2 + f(x)**2), f(x)) == None
      assert homogeneous_order(f(x,y)**2, x, f(x,y), y) == 2
      assert homogeneous_order(f(x,y)**2, x, f(x), y) == None
      assert homogeneous_order(f(x,y)**2, x, f(x,y)) == None
      assert homogeneous_order(f(y,x)**2, x, y, f(x, y)) == None
      assert homogeneous_order(f(y), f(x), x) == None
      assert homogeneous_order(-f(x)/x + 1/sin(f(x)/ x), f(x), x) == 0
      assert homogeneous_order(log(1/y) + log(x**2), x, y) == None
      assert homogeneous_order(log(1/y) + log(x), x, y) == 0
      assert homogeneous_order(log(x/y), x, y) == 0
      assert homogeneous_order(2*log(1/y) + 2*log(x), x, y) == 0
      a = Symbol('a')
      assert homogeneous_order(a*log(1/y) + a*log(x), x, y) == 0
      assert homogeneous_order(f(x).diff(x), x, y) == None
      assert homogeneous_order(-f(x).diff(x) + x, x, y) == None
      assert homogeneous_order(O(x), x, y) == None
      assert homogeneous_order(x + O(x**2), x, y) == None
      assert homogeneous_order(x**pi, x) == pi
      assert homogeneous_order(x**x, x) == None
      raises(ValueError, "homogeneous_order(x*y)")
  
  def test_1st_homogeneous_coeff_ode():
      #skip("These tests pass but take too long.")
      # Type: First order homogeneous, y'=f(y/x)
      eq1 = f(x)/x*cos(f(x)/x) - (x/f(x)*sin(f(x)/x) + cos(f(x)/x))*f(x).diff(x)
      eq2 = x*f(x).diff(x) - f(x) - x*sin(f(x)/x)
      eq3 = f(x) + (x*log(f(x)/x) - 2*x)*diff(f(x),x)
      eq4 = 2*f(x)*exp(x/f(x)) + f(x)*f(x).diff(x) - 2*x*exp(x/f(x))*f(x).diff(x)
      eq5 = 2*x**2*f(x) + f(x)**3 + (x*f(x)**2 - 2*x**3)*f(x).diff(x)
      eq6 = x*exp(f(x)/x) - f(x)*sin(f(x)/x) + x*sin(f(x)/x)*f(x).diff(x)
      eq7 = (x + sqrt(f(x)**2 - x*f(x)))*f(x).diff(x) - f(x)
      eq8 = x+f(x)-(x-f(x))*f(x).diff(x)
      sol1 = Eq(f(x)*sin(f(x)/x), C1)
      sol2 = Eq(x*sqrt(1 + cos(f(x)/x))/sqrt(-1 + cos(f(x)/x)), C1)
      sol3 = Eq(f(x), x*exp(1 - LambertW(C1*x)))
      sol4 = Eq(log(C1*f(x)) + 2*exp(x/f(x)), 0)
      #sol5 = Eq(log(C1*x*sqrt(1/x)*sqrt(f(x))) + x**2/(2*f(x)**2), 0)
      sol5 = Eq(log(C1*x*sqrt(f(x)/x)) + x**2/(2*f(x)**2), 0)
      sol6 = Eq(-exp(-f(x)/x)*sin(f(x)/x)/2 + log(C1*x) -
              cos(f(x)/x)*exp(-f(x)/x)/2, 0)
      sol7 = Eq(log(C1*f(x)) + 2*sqrt(1 - x/f(x)), 0)
      sol8 = Eq(-atan(f(x)/x) + log(C1*x*sqrt(1 + f(x)**2/x**2)), 0)
      assert dsolve(eq1, hint='1st_homogeneous_coeff_subs_dep_div_indep') == sol1
      # indep_div_dep actually has a simpler solution for eq2,
      # but it runs too slow
      assert dsolve(eq2, hint='1st_homogeneous_coeff_subs_dep_div_indep') == sol2
      assert dsolve(eq3, hint='1st_homogeneous_coeff_best') == sol3
      assert dsolve(eq4, hint='1st_homogeneous_coeff_best') == sol4
      assert dsolve(eq5, hint='1st_homogeneous_coeff_best') == sol5
      assert dsolve(eq6, hint='1st_homogeneous_coeff_subs_dep_div_indep') == sol6
      assert dsolve(eq7, hint='1st_homogeneous_coeff_best') == sol7
      assert dsolve(eq8, hint='1st_homogeneous_coeff_best') == sol8
      # checks are below
  
  @slow
  def test_1st_homogeneous_coeff_ode_check14568():
      # These are the checkodesols from test_homogeneous_coeff_ode1.
      eq1 = f(x)/x*cos(f(x)/x) - (x/f(x)*sin(f(x)/x) + cos(f(x)/x))*f(x).diff(x)
      eq4 = 2*f(x)*exp(x/f(x)) + f(x)*f(x).diff(x) - 2*x*exp(x/f(x))*f(x).diff(x)
      eq5 = 2*x**2*f(x) + f(x)**3 + (x*f(x)**2 - 2*x**3)*f(x).diff(x)
      eq6 = x*exp(f(x)/x) - f(x)*sin(f(x)/x) + x*sin(f(x)/x)*f(x).diff(x)
      eq8 = x+f(x)-(x-f(x))*f(x).diff(x)
      sol1 = Eq(f(x)*sin(f(x)/x), C1)
      sol4 = Eq(log(C1*f(x)) + 2*exp(x/f(x)), 0)
      sol5 = Eq(log(C1*x*sqrt(1/x)*sqrt(f(x))) + x**2/(2*f(x)**2), 0)
      sol6 = Eq(-exp(-f(x)/x)*sin(f(x)/x)/2 + log(C1*x) -
              cos(f(x)/x)*exp(-f(x)/x)/2, 0)
      sol8 = Eq(-atan(f(x)/x) + log(C1*x*sqrt(1 + f(x)**2/x**2)), 0)
      assert checkodesol(eq1, sol1, order=1, solve_for_func=False)[0]
      assert checkodesol(eq4, sol4, order=1, solve_for_func=False)[0]
      assert checkodesol(eq5, sol5, order=1, solve_for_func=False)[0]
      assert checkodesol(eq6, sol6, order=1, solve_for_func=False)[0]
      assert checkodesol(eq8, sol8, order=1, solve_for_func=False)[0]
  
  @XFAIL
  def test_1st_homogeneous_coeff_ode_check2():
      skip('This is a known issue.')
      # checker cannot determine that the following expression, z, is
      # zero:
      '''
      >>> checkodesol(eq2, sol2, order=1, solve_for_func=False)
      (False, x*(-sin(f(x)/x) + 2*cos(f(x)/(2*x))**2*tan(f(x)/(2*x))))
      >>> checkodesol(eq2, sol2, order=1, solve_for_func=1)
      (False, x*(-C1**2*sin(2*atan(x/C1)) + 2*C1*x - x**2*sin(2*atan(x/C1))))
      >>> z = _[1]
      >>> z.subs(zip((x,C1),(2,3))).n()
      0.e-124
      >>> z.subs(zip((x,C1),(2,.3))).n()
      4.44089209850063e-16
      >>> z.subs(zip((x,C1),(12,.3))).n()
      0
      '''
      eq2 = x*f(x).diff(x) - f(x) - x*sin(f(x)/x)
      sol2 = Eq(x/tan(f(x)/(2*x)), C1)
      assert checkodesol(eq2, sol2, order=1, solve_for_func=False)[0]
  
  @XFAIL
  def test_1st_homogeneous_coeff_ode_check3():
      skip('This is a known issue.')
      # checker cannot determine that the following expression is zero:
      # (False,
      #   x*(log(exp(-LambertW(C1*x))) +
      #   LambertW(C1*x))*exp(-LambertW(C1*x) + 1))
      eq3 = f(x) + (x*log(f(x)/x) - 2*x)*diff(f(x),x)
      sol3 = Eq(f(x), x*exp(1 - LambertW(C1*x)))
      assert checkodesol(eq3, sol3, solve_for_func=True)[0]
      # and without an assumption about x and f(x), the implicit form doesn't
      # resolve, either:
      # (False, (log(f(x)/x) + log(x/f(x)))*f(x))
      sol3 = Eq(-f(x)/(1 + log(x/f(x))), C1)
      assert checkodesol(eq3, sol3, order=1, solve_for_func=False)[0]
  
  @XFAIL
  def test_1st_homogeneous_coeff_ode_check7():
      skip('This is a known issue.')
      # checker cannot solve for the function and cannot determine that
      # the result is zero:
      '''
      >>> checkodesol(eq7, sol7, order=1, solve_for_func=True)
      (False, (x + sqrt(-x*f(x) + f(x)**2))*(x*f(x) -
      sqrt(-x/f(x) + 1)*f(x)**2)*f(x) -
      (x**2*f(x) - (-x + f(x))*f(x)**2)*f(x))
      '''
      eq7 = (x + sqrt(f(x)**2 - x*f(x)))*f(x).diff(x) - f(x)
      sol7 = Eq(log(C1*f(x)) + 2*sqrt(1 - x/f(x)), 0)
      assert checkodesol(eq7, sol7, order=1, solve_for_func=False)[0]
  
  @XFAIL
  def test_1st_homogeneous_coeff_ode_check9():
      skip('This is a known issue.')
      # checker cannot solve for the function and cannot determine that
      # the result is zero:
      '''
      >>> checkodesol(eq9, sol9, order=1, solve_for_func=True)
      (False, (x*sqrt(-x**2 + f(x)**2) - x*f(x))*
              (x*sqrt(-x**2/f(x)**2 + 1)*f(x) + x*f(x))*f(x)**2 +
              (-x**2*(-x**2 + f(x)**2) + x**2*f(x)**2)*f(x)**2)
      _u2 = Dummy('u2')
      __a = Dummy('a')
      '''
      eq9 = f(x)**2 + (x*sqrt(f(x)**2 - x**2) - x*f(x))*f(x).diff(x)
      sol9 = Eq(-Integral(-1/(-(1 - sqrt(1 - _u2**2))*_u2 + _u2), (_u2, __a,
          x/f(x))) + log(C1*f(x)), 0)
      assert checkodesol(eq9, sol9, order=1, solve_for_func=False)[0]
  
  @XFAIL
  def test_1st_homogeneous_coeff_ode2():
      eq1 = f(x).diff(x) - f(x)/x+1/sin(f(x)/x)
      eq2 = x**2 + f(x)**2 - 2*x*f(x)*f(x).diff(x)
      eq3 = x*exp(f(x)/x) + f(x) - x*f(x).diff(x)
      sol1 = Eq(f(x), x*acos(log(C1*x)))
      sol2 = set([Eq(f(x), -sqrt(C1*x + x**2)), Eq(f(x), sqrt(C1*x + x**2))])
      sol3 = Eq(f(x), log((-1/log(C1*x))**x))
      # specific hints are applied for speed reasons
      assert dsolve(eq1, hint='1st_homogeneous_coeff_subs_dep_div_indep') == sol1
      assert set(dsolve(eq2, hint='1st_homogeneous_coeff_best')) == sol2
      assert dsolve(eq3, hint='1st_homogeneous_coeff_subs_dep_div_indep') == sol3
      assert checkodesol(eq1, sol1, order=1, solve_for_func=False)[0]
      assert all(i[0] for i in
              checkodesol(eq2, sol2, order=1, solve_for_func=False))
      # the solution doesn't check...perhaps there is something wrong with the
      # routine or the solver?
      assert checkodesol(eq3, sol3, order=1, solve_for_func=False)[0]
  
  @XFAIL
  def test_1st_homogeneous_coeff_ode2_check3():
      # simplify() will need to get way better before it can do this one
      eq3 = x*exp(f(x)/x) + f(x) - x*f(x).diff(x)
      sol3 = Eq(f(x), log(log(C1/x)**(-x)))
      assert checkodesol(eq3, sol3, order=1, solve_for_func=False)[0]
  
  def test_1st_homogeneous_coeff_ode3():
      # The standard integration engine cannot handle one of the integrals
      # involved (see issue 1452).  meijerg code comes up with an answer, but in
      # unconventional form.
      # checkodesol fails for this equation, so its test is in
      # test_homogeneous_order_ode1_sol above. It has to compare string
      # expressions because u2 is a dummy variable.
      eq = f(x)**2+(x*sqrt(f(x)**2-x**2)-x*f(x))*f(x).diff(x)
      sol = Eq(Piecewise((-acosh(f(x)/x), 1 < abs(f(x)**2/x**2)),
                         (I*asin(f(x)/x), True)) + log(C1*f(x)), 0)
      assert dsolve(eq, hint='1st_homogeneous_coeff_subs_indep_div_dep') == sol
  
  @XFAIL
  def test_1st_homogeneous_coeff_ode3_check():
      eq = f(x)**2+(x*sqrt(f(x)**2-x**2)-x*f(x))*f(x).diff(x)
      sol = Eq(f(x)**2 - C1*x, f(x)*sqrt(f(x)**2-x**2))
      assert checkodesol(eq, sol, solve_for_func=False)[0]
  
  def test_1st_homogeneous_coeff_corner_case():
      eq1 = f(x).diff(x) - f(x)/x
      c1 = classify_ode(eq1, f(x))
      eq2 = x*f(x).diff(x) - f(x)
      c2 = classify_ode(eq2, f(x))
      sdi = "1st_homogeneous_coeff_subs_dep_div_indep"
      sid = "1st_homogeneous_coeff_subs_indep_div_dep"
      assert sid not in c1 and sdi not in c1
      assert sid not in c2 and sdi not in c2
  
  def test_nth_linear_constant_coeff_homogeneous():
      # From Exercise 20, in Ordinary Differential Equations,
      #                      Tenenbaum and Pollard, pg. 220
      a = Symbol('a', positive=True)
      k = Symbol('k', real=True)
      eq1 = f(x).diff(x, 2) + 2*f(x).diff(x)
      eq2 = f(x).diff(x, 2) - 3*f(x).diff(x) + 2*f(x)
      eq3 = f(x).diff(x, 2) - f(x)
      eq4 = f(x).diff(x, 3) + f(x).diff(x, 2) - 6*f(x).diff(x)
      eq5 = 6*f(x).diff(x, 2) - 11*f(x).diff(x) + 4*f(x)
      eq6 = Eq(f(x).diff(x, 2) + 2*f(x).diff(x) - f(x), 0)
      eq7 = diff(f(x), x, 3) + diff(f(x), x, 2) - 10*diff(f(x), x) - 6*f(x)
      eq8 = f(x).diff(x, 4) - f(x).diff(x, 3) - 4*f(x).diff(x, 2) + \
          4*f(x).diff(x)
      eq9 = f(x).diff(x, 4) + 4*f(x).diff(x, 3) + f(x).diff(x, 2) - \
          4*f(x).diff(x) - 2*f(x)
      eq10 = f(x).diff(x, 4) - a**2*f(x)
      eq11 = f(x).diff(x, 2) - 2*k*f(x).diff(x) - 2*f(x)
      eq12 = f(x).diff(x, 2) + 4*k*f(x).diff(x) - 12*k**2*f(x)
      eq13 = f(x).diff(x, 4)
      eq14 = f(x).diff(x, 2) + 4*f(x).diff(x) + 4*f(x)
      eq15 = 3*f(x).diff(x, 3) + 5*f(x).diff(x, 2) + f(x).diff(x) - f(x)
      eq16 = f(x).diff(x, 3) - 6*f(x).diff(x, 2) +12*f(x).diff(x) - 8*f(x)
      eq17 = f(x).diff(x, 2) - 2*a*f(x).diff(x) + a**2*f(x)
      eq18 = f(x).diff(x, 4) + 3*f(x).diff(x, 3)
      eq19 = f(x).diff(x, 4) - 2*f(x).diff(x, 2)
      eq20 = f(x).diff(x, 4) + 2*f(x).diff(x, 3) - 11*f(x).diff(x, 2) - \
          12*f(x).diff(x) + 36*f(x)
      eq21 = 36*f(x).diff(x, 4) - 37*f(x).diff(x, 2) + 4*f(x).diff(x) + 5*f(x)
      eq22 = f(x).diff(x, 4) - 8*f(x).diff(x, 2) + 16*f(x)
      eq23 = f(x).diff(x, 2) - 2*f(x).diff(x) + 5*f(x)
      eq24 = f(x).diff(x, 2) - f(x).diff(x) + f(x)
      eq25 = f(x).diff(x, 4) + 5*f(x).diff(x, 2) + 6*f(x)
      eq26 = f(x).diff(x, 2) - 4*f(x).diff(x) + 20*f(x)
      eq27 = f(x).diff(x, 4) + 4*f(x).diff(x, 2) + 4*f(x)
      eq28 = f(x).diff(x, 3) + 8*f(x)
      eq29 = f(x).diff(x, 4) + 4*f(x).diff(x, 2)
      eq30 = f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x)
      sol1 = Eq(f(x), C1 + C2*exp(-2*x))
      sol2 = Eq(f(x), (C1*exp(x) + C2*exp(2*x)))
      sol3 = Eq(f(x), C1*exp(x) + C2*exp(-x))
      sol4 = Eq(f(x), C1 + C2*exp(-3*x) + C3*exp(2*x))
      sol5 = Eq(f(x), C1*exp(x/2) + C2*exp(4*x/3))
      sol6 = Eq(f(x), C1*exp(x*(-1 + sqrt(2))) + C2*exp(x*(-sqrt(2) - 1)))
      sol7 = Eq(f(x),
          C1*exp(3*x) + C2*exp(x*(-2 - sqrt(2))) + C3*exp(x*(-2 + sqrt(2))))
      sol8 = Eq(f(x), C1 + C2*exp(x) + C3*exp(-2*x) + C4*exp(2*x))
      sol9 = Eq(f(x),
          C1*exp(x) + C2*exp(-x) + C3*exp(x*(-2 + sqrt(2))) +
          C4*exp(x*(-2 - sqrt(2))))
      sol10 = Eq(f(x),
          C1*sin(x*sqrt(a)) + C2*cos(x*sqrt(a)) + C3*exp(x*sqrt(a)) +
          C4*exp(-x*sqrt(a)))
      sol11 = Eq(f(x),
          C1*exp(x*(k - sqrt(k**2 + 2))) + C2*exp(x*(k + sqrt(k**2 + 2))))
      sol12 = Eq(f(x),
          C1*exp(2*x*(-2*abs(k) - k)) + C2*exp(2*x*(2*abs(k) - k)))
      sol13 = Eq(f(x), C1 + C2*x + C3*x**2 + C4*x**3)
      sol14 = Eq(f(x), (C1 + C2*x)*exp(-2*x))
      sol15 = Eq(f(x), (C1 + C2*x)*exp(-x) + C3*exp(x/3))
      sol16 = Eq(f(x), (C1 + C2*x + C3*x**2)*exp(2*x))
      sol17 = Eq(f(x), (C1 + C2*x)*exp(a*x))
      sol18 = Eq(f(x), C1 + C2*x + C3*x**2 + C4*exp(-3*x))
      sol19 = Eq(f(x), C1 + C2*x + C3*exp(x*sqrt(2)) + C4*exp(-x*sqrt(2)))
      sol20 = Eq(f(x), (C1 + C2*x)*exp(-3*x) + (C3 + C4*x)*exp(2*x))
      sol21 = Eq(f(x), C1*exp(x/2) + C2*exp(-x) + C3*exp(-x/3) + C4*exp(5*x/6))
      sol22 = Eq(f(x), (C1 + C2*x)*exp(-2*x) + (C3 + C4*x)*exp(2*x))
      sol23 = Eq(f(x), (C1*sin(2*x) + C2*cos(2*x))*exp(x))
      sol24 = Eq(f(x), (C1*sin(x*sqrt(3)/2) + C2*cos(x*sqrt(3)/2))*exp(x/2))
      sol25 = Eq(f(x),
          C1*cos(x*sqrt(3)) + C2*sin(x*sqrt(3)) + C3*sin(x*sqrt(2)) +
          C4*cos(x*sqrt(2)))
      sol26 = Eq(f(x), (C1*sin(4*x) + C2*cos(4*x))*exp(2*x))
      sol27 = Eq(f(x), (C1 + C2*x)*sin(x*sqrt(2)) + (C3 + C4*x)*cos(x*sqrt(2)))
      sol28 = Eq(f(x),
          (C1*sin(x*sqrt(3)) + C2*cos(x*sqrt(3)))*exp(x) + C3*exp(-2*x))
      sol29 = Eq(f(x), C1 + C2*sin(2*x) + C3*cos(2*x) + C4*x)
      sol30 = Eq(f(x), C1 + (C2 + C3*x)*sin(x) + (C4 + C5*x)*cos(x))
      sol1s = constant_renumber(sol1, 'C', 1, 2)
      sol2s = constant_renumber(sol2, 'C', 1, 2)
      sol3s = constant_renumber(sol3, 'C', 1, 2)
      sol4s = constant_renumber(sol4, 'C', 1, 3)
      sol5s = constant_renumber(sol5, 'C', 1, 2)
      sol6s = constant_renumber(sol6, 'C', 1, 2)
      sol7s = constant_renumber(sol7, 'C', 1, 3)
      sol8s = constant_renumber(sol8, 'C', 1, 4)
      sol9s = constant_renumber(sol9, 'C', 1, 4)
      sol10s = constant_renumber(sol10, 'C', 1, 4)
      sol11s = constant_renumber(sol11, 'C', 1, 2)
      sol12s = constant_renumber(sol12, 'C', 1, 2)
      sol13s = constant_renumber(sol13, 'C', 1, 4)
      sol14s = constant_renumber(sol14, 'C', 1, 2)
      sol15s = constant_renumber(sol15, 'C', 1, 3)
      sol16s = constant_renumber(sol16, 'C', 1, 3)
      sol17s = constant_renumber(sol17, 'C', 1, 2)
      sol18s = constant_renumber(sol18, 'C', 1, 4)
      sol19s = constant_renumber(sol19, 'C', 1, 4)
      sol20s = constant_renumber(sol20, 'C', 1, 4)
      sol21s = constant_renumber(sol21, 'C', 1, 4)
      sol22s = constant_renumber(sol22, 'C', 1, 4)
      sol23s = constant_renumber(sol23, 'C', 1, 2)
      sol24s = constant_renumber(sol24, 'C', 1, 2)
      sol25s = constant_renumber(sol25, 'C', 1, 4)
      sol26s = constant_renumber(sol26, 'C', 1, 2)
      sol27s = constant_renumber(sol27, 'C', 1, 4)
      sol28s = constant_renumber(sol28, 'C', 1, 3)
      sol29s = constant_renumber(sol29, 'C', 1, 4)
      sol30s = constant_renumber(sol30, 'C', 1, 5)
      assert dsolve(eq1) in (sol1, sol1s)
      assert dsolve(eq2) in (sol2, sol2s)
      assert dsolve(eq3) in (sol3, sol3s)
      assert dsolve(eq4) in (sol4, sol4s)
      assert dsolve(eq5) in (sol5, sol5s)
      assert dsolve(eq6) in (sol6, sol6s)
      assert dsolve(eq7) in (sol7, sol7s)
      assert dsolve(eq8) in (sol8, sol8s)
      assert dsolve(eq9) in (sol9, sol9s)
      assert dsolve(eq10) in (sol10, sol10s)
      assert dsolve(eq11) in (sol11, sol11s)
      assert dsolve(eq12) in (sol12, sol12s)
      assert dsolve(eq13) in (sol13, sol13s)
      assert dsolve(eq14) in (sol14, sol14s)
      assert dsolve(eq15) in (sol15, sol15s)
      assert dsolve(eq16) in (sol16, sol16s)
      assert dsolve(eq17) in (sol17, sol17s)
      assert dsolve(eq18) in (sol18, sol18s)
      assert dsolve(eq19) in (sol19, sol19s)
      assert dsolve(eq20) in (sol20, sol20s)
      assert dsolve(eq21) in (sol21, sol21s)
      assert dsolve(eq22) in (sol22, sol22s)
      assert dsolve(eq23) in (sol23, sol23s)
      assert dsolve(eq24) in (sol24, sol24s)
      assert dsolve(eq25) in (sol25, sol25s)
      assert dsolve(eq26) in (sol26, sol26s)
      assert dsolve(eq27) in (sol27, sol27s)
      assert dsolve(eq28) in (sol28, sol28s)
      assert dsolve(eq29) in (sol29, sol29s)
      assert dsolve(eq30) in (sol30, sol30s)
      assert checkodesol(eq1, sol1, order=2, solve_for_func=False)[0]
      assert checkodesol(eq2, sol2, order=2, solve_for_func=False)[0]
      assert checkodesol(eq3, sol3, order=2, solve_for_func=False)[0]
      assert checkodesol(eq4, sol4, order=3, solve_for_func=False)[0]
      assert checkodesol(eq5, sol5, order=2, solve_for_func=False)[0]
      assert checkodesol(eq6, sol6, order=2, solve_for_func=False)[0]
      assert checkodesol(eq7, sol7, order=3, solve_for_func=False)[0]
      assert checkodesol(eq8, sol8, order=4, solve_for_func=False)[0]
      assert checkodesol(eq9, sol9, order=4, solve_for_func=False)[0]
      assert checkodesol(eq10, sol10, order=4, solve_for_func=False)[0]
      assert checkodesol(eq11, sol11, order=2, solve_for_func=False)[0]
      assert checkodesol(eq12, sol12, order=2, solve_for_func=False)[0]
      assert checkodesol(eq13, sol13, order=4, solve_for_func=False)[0]
      assert checkodesol(eq14, sol14, order=2, solve_for_func=False)[0]
      assert checkodesol(eq15, sol15, order=3, solve_for_func=False)[0]
      assert checkodesol(eq16, sol16, order=3, solve_for_func=False)[0]
      assert checkodesol(eq17, sol17, order=2, solve_for_func=False)[0]
      assert checkodesol(eq18, sol18, order=4, solve_for_func=False)[0]
      assert checkodesol(eq19, sol19, order=4, solve_for_func=False)[0]
      assert checkodesol(eq20, sol20, order=4, solve_for_func=False)[0]
      assert checkodesol(eq21, sol21, order=4, solve_for_func=False)[0]
      assert checkodesol(eq22, sol22, order=4, solve_for_func=False)[0]
      assert checkodesol(eq23, sol23, order=2, solve_for_func=False)[0]
      assert checkodesol(eq24, sol24, order=2, solve_for_func=False)[0]
      assert checkodesol(eq25, sol25, order=4, solve_for_func=False)[0]
      assert checkodesol(eq26, sol26, order=2, solve_for_func=False)[0]
      assert checkodesol(eq27, sol27, order=4, solve_for_func=False)[0]
      assert checkodesol(eq28, sol28, order=3, solve_for_func=False)[0]
      assert checkodesol(eq29, sol29, order=4, solve_for_func=False)[0]
      assert checkodesol(eq30, sol30, order=5, solve_for_func=False)[0]
  
  def test_nth_linear_constant_coeff_homogeneous_RootOf():
      eq = f(x).diff(x, 5) + 11*f(x).diff(x) - 2*f(x)
      sol = Eq(f(x),
          C1*exp(x*RootOf(x**5 + 11*x - 2, 0)) + \
          C2*exp(x*RootOf(x**5 + 11*x - 2, 1)) + \
          C3*exp(x*RootOf(x**5 + 11*x - 2, 2)) + \
          C4*exp(x*RootOf(x**5 + 11*x - 2, 3)) + \
          C5*exp(x*RootOf(x**5 + 11*x - 2, 4)))
      assert dsolve(eq) == sol
  
  @XFAIL
  def test_nth_linear_constant_coeff_homogeneous_RootOf_sol():
      eq = f(x).diff(x, 5) + 11*f(x).diff(x) - 2*f(x)
      sol = Eq(f(x),
          C1*exp(x*RootOf(x**5 + 11*x - 2, 0)) + \
          C2*exp(x*RootOf(x**5 + 11*x - 2, 1)) + \
          C3*exp(x*RootOf(x**5 + 11*x - 2, 2)) + \
          C4*exp(x*RootOf(x**5 + 11*x - 2, 3)) + \
          C5*exp(x*RootOf(x**5 + 11*x - 2, 4)))
      assert checkodesol(eq, sol, order=5, solve_for_func=False)[0]
  
  def test_undetermined_coefficients_match():
      assert _undetermined_coefficients_match(g(x), x) == {'test': False}
      assert _undetermined_coefficients_match(sin(2*x + sqrt(5)), x) == \
          {'test': True, 'trialset':
              set([cos(2*x + sqrt(5)), sin(2*x + sqrt(5))])}
      assert _undetermined_coefficients_match(sin(x)*cos(x), x) == \
          {'test': False}
      s = set([cos(x), x*cos(x), x**2*cos(x), x**2*sin(x), x*sin(x), sin(x)])
      assert _undetermined_coefficients_match(sin(x)*(x**2 + x + 1), x) == \
          {'test': True, 'trialset': s}
      assert _undetermined_coefficients_match(
          sin(x)*x**2 + sin(x)*x + sin(x), x) == {'test': True, 'trialset': s}
      assert _undetermined_coefficients_match(
          exp(2*x)*sin(x)*(x**2 + x + 1), x
          ) == {
          'test': True, 'trialset': set([exp(2*x)*sin(x), x**2*exp(2*x)*sin(x),
          cos(x)*exp(2*x), x**2*cos(x)*exp(2*x), x*cos(x)*exp(2*x),
          x*exp(2*x)*sin(x)])}
      assert _undetermined_coefficients_match(1/sin(x), x) == {'test': False}
      assert _undetermined_coefficients_match(log(x), x) == {'test': False}
      assert _undetermined_coefficients_match(2**(x)*(x**2 + x + 1), x) == \
          {'test': True, 'trialset': set([2**x, x*2**x, x**2*2**x])}
      assert _undetermined_coefficients_match(x**y, x) == {'test': False}
      assert _undetermined_coefficients_match(exp(x)*exp(2*x + 1), x) == \
          {'test': True, 'trialset': set([exp(1 + 3*x)])}
      assert _undetermined_coefficients_match(sin(x)*(x**2 + x + 1), x) == \
          {'test': True, 'trialset': set([x*cos(x), x*sin(x), x**2*cos(x),
          x**2*sin(x), cos(x), sin(x)])}
      assert _undetermined_coefficients_match(sin(x)*(x + sin(x)), x) == \
          {'test': False}
      assert _undetermined_coefficients_match(sin(x)*(x + sin(2*x)), x) == \
          {'test': False}
      assert _undetermined_coefficients_match(sin(x)*tan(x), x) == \
          {'test': False}
      assert _undetermined_coefficients_match(
          x**2*sin(x)*exp(x) + x*sin(x) + x, x
          ) == {
          'test': True, 'trialset': set([x**2*cos(x)*exp(x), x, cos(x), S(1),
          exp(x)*sin(x), sin(x), x*exp(x)*sin(x), x*cos(x), x*cos(x)*exp(x),
          x*sin(x), cos(x)*exp(x), x**2*exp(x)*sin(x)])}
      assert _undetermined_coefficients_match(4*x*sin(x - 2), x) == \
          {'test': True, 'trialset': set([x*cos(x - 2), x*sin(x - 2), cos(x - 2),
          sin(x - 2)])}
      assert _undetermined_coefficients_match(2**x*x, x) == \
          {'test': True, 'trialset': set([2**x, x*2**x])}
      assert _undetermined_coefficients_match(2**x*exp(2*x), x) == \
          {'test': True, 'trialset': set([2**x*exp(2*x)])}
      assert _undetermined_coefficients_match(exp(-x)/x, x) == \
          {'test': False}
      # Below are from Ordinary Differential Equations,
      #                Tenenbaum and Pollard, pg. 231
      assert _undetermined_coefficients_match(S(4), x) == \
          {'test': True, 'trialset': set([S(1)])}
      assert _undetermined_coefficients_match(12*exp(x), x) == \
          {'test': True, 'trialset': set([exp(x)])}
      assert _undetermined_coefficients_match(exp(I*x), x) == \
          {'test': True, 'trialset': set([exp(I*x)])}
      assert _undetermined_coefficients_match(sin(x), x) == \
          {'test': True, 'trialset': set([cos(x), sin(x)])}
      assert _undetermined_coefficients_match(cos(x), x) == \
          {'test': True, 'trialset': set([cos(x), sin(x)])}
      assert _undetermined_coefficients_match(8 + 6*exp(x) + 2*sin(x), x) == \
          {'test': True, 'trialset': set([S(1), cos(x), sin(x), exp(x)])}
      assert _undetermined_coefficients_match(x**2, x) == \
          {'test': True, 'trialset': set([S(1), x, x**2])}
      assert _undetermined_coefficients_match(9*x*exp(x) + exp(-x), x) == \
          {'test': True, 'trialset': set([x*exp(x), exp(x), exp(-x)])}
      assert _undetermined_coefficients_match(2*exp(2*x)*sin(x), x) == \
          {'test': True, 'trialset': set([exp(2*x)*sin(x), cos(x)*exp(2*x)])}
      assert _undetermined_coefficients_match(x - sin(x), x) == \
          {'test': True, 'trialset': set([S(1), x, cos(x), sin(x)])}
      assert _undetermined_coefficients_match(x**2 + 2*x, x) == \
          {'test': True, 'trialset': set([S(1), x, x**2])}
      assert _undetermined_coefficients_match(4*x*sin(x), x) == \
          {'test': True, 'trialset': set([x*cos(x), x*sin(x), cos(x), sin(x)])}
      assert _undetermined_coefficients_match(x*sin(2*x), x) == \
          {'test': True, 'trialset':
              set([x*cos(2*x), x*sin(2*x), cos(2*x), sin(2*x)])}
      assert _undetermined_coefficients_match(x**2*exp(-x), x) == \
          {'test': True, 'trialset': set([x*exp(-x), x**2*exp(-x), exp(-x)])}
      assert _undetermined_coefficients_match(2*exp(-x) - x**2*exp(-x), x) == \
          {'test': True, 'trialset': set([x*exp(-x), x**2*exp(-x), exp(-x)])}
      assert _undetermined_coefficients_match(exp(-2*x) + x**2, x) == \
          {'test': True, 'trialset': set([S(1), x, x**2, exp(-2*x)])}
      assert _undetermined_coefficients_match(x*exp(-x), x) == \
          {'test': True, 'trialset': set([x*exp(-x), exp(-x)])}
      assert _undetermined_coefficients_match(x + exp(2*x), x) == \
          {'test': True, 'trialset': set([S(1), x, exp(2*x)])}
      assert _undetermined_coefficients_match(sin(x) + exp(-x), x) == \
          {'test': True, 'trialset': set([cos(x), sin(x), exp(-x)])}
      assert _undetermined_coefficients_match(exp(x), x) == \
          {'test': True, 'trialset': set([exp(x)])}
      # converted from sin(x)**2
      assert _undetermined_coefficients_match(S(1)/2 - cos(2*x)/2, x) == \
      {'test': True, 'trialset': set([S(1), cos(2*x), sin(2*x)])}
      # converted from exp(2*x)*sin(x)**2
      assert _undetermined_coefficients_match(
          exp(2*x)*(S(1)/2 + cos(2*x)/2), x
          ) == {
          'test': True, 'trialset': set([exp(2*x)*sin(2*x), cos(2*x)*exp(2*x),
          exp(2*x)])}
      assert _undetermined_coefficients_match(2*x + sin(x) + cos(x), x) == \
          {'test': True, 'trialset': set([S(1), x, cos(x), sin(x)])}
      # converted from sin(2*x)*sin(x)
      assert _undetermined_coefficients_match(cos(x)/2 - cos(3*x)/2, x) == \
          {'test': True, 'trialset': set([cos(x), cos(3*x), sin(x), sin(3*x)])}
      assert _undetermined_coefficients_match(cos(x**2), x) == {'test': False}
      assert _undetermined_coefficients_match(2**(x**2), x) == {'test': False}
  
  def test_nth_linear_constant_coeff_undetermined_coefficients():
      hint = 'nth_linear_constant_coeff_undetermined_coefficients'
      g = exp(-x)
      f2 = f(x).diff(x, 2)
      c = 3*f(x).diff(x, 3) + 5*f2 + f(x).diff(x) - f(x) - x
      eq1 = c - x*g
      eq2 = c - g
      # 3-27 below are from Ordinary Differential Equations,
      #                     Tenenbaum and Pollard, pg. 231
      eq3 = f2 + 3*f(x).diff(x) + 2*f(x) - 4
      eq4 = f2 + 3*f(x).diff(x) + 2*f(x) - 12*exp(x)
      eq5 = f2 + 3*f(x).diff(x) + 2*f(x) - exp(I*x)
      eq6 = f2 + 3*f(x).diff(x) + 2*f(x) - sin(x)
      eq7 = f2 + 3*f(x).diff(x) + 2*f(x) - cos(x)
      eq8 = f2 + 3*f(x).diff(x) + 2*f(x) - (8 + 6*exp(x) + 2*sin(x))
      eq9 = f2 + f(x).diff(x) + f(x) - x**2
      eq10 = f2 - 2*f(x).diff(x) - 8*f(x) - 9*x*exp(x) - 10*exp(-x)
      eq11 = f2 - 3*f(x).diff(x) - 2*exp(2*x)*sin(x)
      eq12 = f(x).diff(x, 4) - 2*f2 + f(x) - x + sin(x)
      eq13 = f2 + f(x).diff(x) - x**2 - 2*x
      eq14 = f2 + f(x).diff(x) - x - sin(2*x)
      eq15 = f2 + f(x) - 4*x*sin(x)
      eq16 = f2 + 4*f(x) - x*sin(2*x)
      eq17 = f2 + 2*f(x).diff(x) + f(x) - x**2*exp(-x)
      eq18 = f(x).diff(x, 3) + 3*f2 + 3*f(x).diff(x) + f(x) - 2*exp(-x) + \
          x**2*exp(-x)
      eq19 = f2 + 3*f(x).diff(x) + 2*f(x) - exp(-2*x) - x**2
      eq20 = f2 - 3*f(x).diff(x) + 2*f(x) - x*exp(-x)
      eq21 = f2 + f(x).diff(x) - 6*f(x) - x - exp(2*x)
      eq22 = f2 + f(x) - sin(x) - exp(-x)
      eq23 = f(x).diff(x, 3) - 3*f2 + 3*f(x).diff(x) - f(x) - exp(x)
      # sin(x)**2
      eq24 = f2 + f(x) - S(1)/2 - cos(2*x)/2
      # exp(2*x)*sin(x)**2
      eq25 = f(x).diff(x, 3) - f(x).diff(x) - exp(2*x)*(S(1)/2 - cos(2*x)/2)
      eq26 = (f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x) - 2*x -
          sin(x) - cos(x))
      # sin(2*x)*sin(x), skip 3127 for now, match bug
      eq27 = f2 + f(x) - cos(x)/2 + cos(3*x)/2
      eq28 = f(x).diff(x) - 1
      sol1 = Eq(f(x),
          -1 - x + (C1 + C2*x - 3*x**2/32 - x**3/24)*exp(-x) + C3*exp(x/3))
      sol2 = Eq(f(x), -1 - x + (C1 + C2*x - x**2/8)*exp(-x) + C3*exp(x/3))
      sol3 = Eq(f(x), 2 + C1*exp(-x) + C2*exp(-2*x))
      sol4 = Eq(f(x), 2*exp(x) + C1*exp(-x) + C2*exp(-2*x))
      sol5 = Eq(f(x), C1*exp(-x) + C2*exp(-2*x) + (S(1)/10 - 3*I/10)*exp(I*x))
      sol6 = Eq(f(x), -3*cos(x)/10 + sin(x)/10 + C1*exp(-x) + C2*exp(-2*x))
      sol7 = Eq(f(x), cos(x)/10 + 3*sin(x)/10 + C1*exp(-x) + C2*exp(-2*x))
      sol8 = Eq(f(x),
          4 - 3*cos(x)/5 + sin(x)/5 + exp(x) + C1*exp(-x) + C2*exp(-2*x))
      sol9 = Eq(f(x),
          -2*x + x**2 + (C1*sin(x*sqrt(3)/2) + C2*cos(x*sqrt(3)/2))*exp(-x/2))
      sol10 = Eq(f(x), -x*exp(x) - 2*exp(-x) + C1*exp(-2*x) + C2*exp(4*x))
      sol11 = Eq(f(x), C1 + C2*exp(3*x) + (-3*sin(x) - cos(x))*exp(2*x)/5)
      sol12 = Eq(f(x), x - sin(x)/4 + (C1 + C2*x)*exp(-x) + (C3 + C4*x)*exp(x))
      sol13 = Eq(f(x), C1 + x**3/3 + C2*exp(-x))
      sol14 = Eq(f(x), C1 - x - sin(2*x)/5 - cos(2*x)/10 + x**2/2 + C2*exp(-x))
      sol15 = Eq(f(x), (C1 + x)*sin(x) + (C2 - x**2)*cos(x))
      sol16 = Eq(f(x), (C1 + x/16)*sin(2*x) + (C2 - x**2/8)*cos(2*x))
      sol17 = Eq(f(x), (C1 + C2*x + x**4/12)*exp(-x))
      sol18 = Eq(f(x), (C1 + C2*x + C3*x**2 - x**5/60 + x**3/3)*exp(-x))
      sol19 = Eq(f(x), S(7)/4 - 3*x/2 + x**2/2 + C1*exp(-x) + (C2 - x)*exp(-2*x))
      sol20 = Eq(f(x), C1*exp(x) + C2*exp(2*x) + (6*x + 5)*exp(-x)/36)
      sol21 = Eq(f(x), -S(1)/36 - x/6 + C1*exp(-3*x) + (C2 + x/5)*exp(2*x))
      sol22 = Eq(f(x), C1*sin(x) + (C2 - x/2)*cos(x) + exp(-x)/2)
      sol23 = Eq(f(x), (C1 + C2*x + C3*x**2 + x**3/6)*exp(x))
      sol24 = Eq(f(x), S(1)/2 - cos(2*x)/6 + C1*sin(x) + C2*cos(x))
      sol25 = Eq(f(x), C1 + C2*exp(-x) + C3*exp(x) + \
                      (-21*sin(2*x) + 27*cos(2*x) + 130)*exp(2*x)/1560)
      sol26 = Eq(f(x),
          C1 + (C2 + C3*x - x**2/8)*sin(x) + (C4 + C5*x + x**2/8)*cos(x) + x**2)
      sol27 = Eq(f(x), cos(3*x)/16 + C1*cos(x) + (C2 + x/4)*sin(x))
      sol28 = Eq(f(x), C1 + x)
      sol1s = constant_renumber(sol1, 'C', 1, 3)
      sol2s = constant_renumber(sol2, 'C', 1, 3)
      sol3s = constant_renumber(sol3, 'C', 1, 2)
      sol4s = constant_renumber(sol4, 'C', 1, 2)
      sol5s = constant_renumber(sol5, 'C', 1, 2)
      sol6s = constant_renumber(sol6, 'C', 1, 2)
      sol7s = constant_renumber(sol7, 'C', 1, 2)
      sol8s = constant_renumber(sol8, 'C', 1, 2)
      sol9s = constant_renumber(sol9, 'C', 1, 2)
      sol10s = constant_renumber(sol10, 'C', 1, 2)
      sol11s = constant_renumber(sol11, 'C', 1, 2)
      sol12s = constant_renumber(sol12, 'C', 1, 2)
      sol13s = constant_renumber(sol13, 'C', 1, 4)
      sol14s = constant_renumber(sol14, 'C', 1, 2)
      sol15s = constant_renumber(sol15, 'C', 1, 2)
      sol16s = constant_renumber(sol16, 'C', 1, 2)
      sol17s = constant_renumber(sol17, 'C', 1, 2)
      sol18s = constant_renumber(sol18, 'C', 1, 3)
      sol19s = constant_renumber(sol19, 'C', 1, 2)
      sol20s = constant_renumber(sol20, 'C', 1, 2)
      sol21s = constant_renumber(sol21, 'C', 1, 2)
      sol22s = constant_renumber(sol22, 'C', 1, 2)
      sol23s = constant_renumber(sol23, 'C', 1, 3)
      sol24s = constant_renumber(sol24, 'C', 1, 2)
      sol25s = constant_renumber(sol25, 'C', 1, 3)
      sol26s = constant_renumber(sol26, 'C', 1, 5)
      sol27s = constant_renumber(sol27, 'C', 1, 2)
      assert dsolve(eq1, hint=hint) in (sol1, sol1s)
      assert dsolve(eq2, hint=hint) in (sol2, sol2s)
      assert dsolve(eq3, hint=hint) in (sol3, sol3s)
      assert dsolve(eq4, hint=hint) in (sol4, sol4s)
      assert dsolve(eq5, hint=hint) in (sol5, sol5s)
      assert dsolve(eq6, hint=hint) in (sol6, sol6s)
      assert dsolve(eq7, hint=hint) in (sol7, sol7s)
      assert dsolve(eq8, hint=hint) in (sol8, sol8s)
      assert dsolve(eq9, hint=hint) in (sol9, sol9s)
      assert dsolve(eq10, hint=hint) in (sol10, sol10s)
      assert dsolve(eq11, hint=hint) in (sol11, sol11s)
      assert dsolve(eq12, hint=hint) in (sol12, sol12s)
      assert dsolve(eq13, hint=hint) in (sol13, sol13s)
      assert dsolve(eq14, hint=hint) in (sol14, sol14s)
      assert dsolve(eq15, hint=hint) in (sol15, sol15s)
      assert dsolve(eq16, hint=hint) in (sol16, sol16s)
      assert dsolve(eq17, hint=hint) in (sol17, sol17s)
      assert dsolve(eq18, hint=hint) in (sol18, sol18s)
      assert dsolve(eq19, hint=hint) in (sol19, sol19s)
      assert dsolve(eq20, hint=hint) in (sol20, sol20s)
      assert dsolve(eq21, hint=hint) in (sol21, sol21s)
      assert dsolve(eq22, hint=hint) in (sol22, sol22s)
      assert dsolve(eq23, hint=hint) in (sol23, sol23s)
      assert dsolve(eq24, hint=hint) in (sol24, sol24s)
      assert dsolve(eq25, hint=hint) in (sol25, sol25s)
      assert dsolve(eq26, hint=hint) in (sol26, sol26s)
      assert dsolve(eq27, hint=hint) in (sol27, sol27s)
      assert dsolve(eq28, hint=hint) == sol28
      assert checkodesol(eq1, sol1, order=3, solve_for_func=False)[0]
      assert checkodesol(eq2, sol2, order=3, solve_for_func=False)[0]
      assert checkodesol(eq3, sol3, order=2, solve_for_func=False)[0]
      assert checkodesol(eq4, sol4, order=2, solve_for_func=False)[0]
      assert checkodesol(eq5, sol5, order=2, solve_for_func=False)[0]
      assert checkodesol(eq6, sol6, order=2, solve_for_func=False)[0]
      assert checkodesol(eq7, sol7, order=2, solve_for_func=False)[0]
      assert checkodesol(eq8, sol8, order=2, solve_for_func=False)[0]
      assert checkodesol(eq9, sol9, order=2, solve_for_func=False)[0]
      assert checkodesol(eq10, sol10, order=2, solve_for_func=False)[0]
      assert checkodesol(eq11, sol11, order=2, solve_for_func=False)[0]
      assert checkodesol(eq12, sol12, order=4, solve_for_func=False)[0]
      assert checkodesol(eq13, sol13, order=2, solve_for_func=False)[0]
      assert checkodesol(eq14, sol14, order=2, solve_for_func=False)[0]
      assert checkodesol(eq15, sol15, order=2, solve_for_func=False)[0]
      assert checkodesol(eq16, sol16, order=2, solve_for_func=False)[0]
      assert checkodesol(eq17, sol17, order=2, solve_for_func=False)[0]
      assert checkodesol(eq18, sol18, order=3, solve_for_func=False)[0]
      assert checkodesol(eq19, sol19, order=2, solve_for_func=False)[0]
      assert checkodesol(eq20, sol20, order=2, solve_for_func=False)[0]
      assert checkodesol(eq21, sol21, order=2, solve_for_func=False)[0]
      assert checkodesol(eq22, sol22, order=2, solve_for_func=False)[0]
      assert checkodesol(eq23, sol23, order=3, solve_for_func=False)[0]
      assert checkodesol(eq24, sol24, order=2, solve_for_func=False)[0]
      assert checkodesol(eq25, sol25, order=3, solve_for_func=False)[0]
      assert checkodesol(eq26, sol26, order=5, solve_for_func=False)[0]
      assert checkodesol(eq27, sol27, order=2, solve_for_func=False)[0]
      assert checkodesol(eq28, sol28, order=1, solve_for_func=False)[0]
  
  @XFAIL
  def test_nth_linear_constant_coeff_undetermined_coefficients_imaginary_exp():
      # Equivalent to eq26 in
      # test_nth_linear_constant_coeff_undetermined_coefficients above.
      # This fails because the algorithm for undetermined coefficients
      # doesn't know to multiply exp(I*x) by sufficient x because it is linearly
      # dependent of sin(x) and cos(x).
      eq26a = f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x) - 2*x - exp(I*x)
      sol26 = Eq(f(x),
          C1 + (C2 + C3*x - x**2/8)*sin(x) + (C4 + C5*x + x**2/8)*cos(x) + x**2)
      assert dsolve(eq26a, hint=hint) == sol26
      assert checkodesol(eq26a, sol26, order=5, solve_for_func=False)[0]
  
  def test_nth_linear_constant_coeff_variation_of_parameters():
      hint = 'nth_linear_constant_coeff_variation_of_parameters'
      g = exp(-x)
      f2 = f(x).diff(x, 2)
      c = 3*f(x).diff(x, 3) + 5*f2 + f(x).diff(x) - f(x) - x
      eq1 = c - x*g
      eq2 = c - g
      eq3 = f(x).diff(x) - 1
      eq4 = f2 + 3*f(x).diff(x) + 2*f(x) - 4
      eq5 = f2 + 3*f(x).diff(x) + 2*f(x) - 12*exp(x)
      eq6 = f2 - 2*f(x).diff(x) - 8*f(x) - 9*x*exp(x) - 10*exp(-x)
      eq7 = f2 + 2*f(x).diff(x) + f(x) - x**2*exp(-x)
      eq8 = f2 - 3*f(x).diff(x) + 2*f(x) - x*exp(-x)
      eq9 = f(x).diff(x, 3) - 3*f2 + 3*f(x).diff(x) - f(x) - exp(x)
      eq10 = f2 + 2*f(x).diff(x) + f(x) - exp(-x)/x
      eq11 = f2 + f(x) - 1/sin(x)*1/cos(x)
      eq12 = f(x).diff(x, 4)  - 1/x
      sol1 = Eq(f(x),
          -1 - x + (C1 + C2*x - 3*x**2/32 - x**3/24)*exp(-x) + C3*exp(x/3))
      sol2 = Eq(f(x), -1 - x + (C1 + C2*x - x**2/8)*exp(-x) + C3*exp(x/3))
      sol3 = Eq(f(x), C1 + x)
      sol4 = Eq(f(x), 2 + C1*exp(-x) + C2*exp(-2*x))
      sol5 = Eq(f(x), 2*exp(x) + C1*exp(-x) + C2*exp(-2*x))
      sol6 = Eq(f(x), -x*exp(x) - 2*exp(-x) + C1*exp(-2*x) + C2*exp(4*x))
      sol7 = Eq(f(x), (C1 + C2*x + x**4/12)*exp(-x))
      sol8 = Eq(f(x), C1*exp(x) + C2*exp(2*x) + (6*x + 5)*exp(-x)/36)
      sol9 = Eq(f(x), (C1 + C2*x + C3*x**2 + x**3/6)*exp(x))
      sol10 = Eq(f(x), (C1 + x*(C2 + log(x)))*exp(-x))
      sol11 = Eq(f(x), cos(x)*(C2 - Integral(1/cos(x), x)) + sin(x)*(C1 + \
          Integral(1/sin(x), x)))
      sol12 = Eq(f(x), C1 + C2*x + x**3*(C3 + log(x)/6) + C4*x**2)
      sol1s = constant_renumber(sol1, 'C', 1, 3)
      sol2s = constant_renumber(sol2, 'C', 1, 3)
      sol3s = constant_renumber(sol3, 'C', 1, 2)
      sol4s = constant_renumber(sol4, 'C', 1, 2)
      sol5s = constant_renumber(sol5, 'C', 1, 2)
      sol6s = constant_renumber(sol6, 'C', 1, 2)
      sol7s = constant_renumber(sol7, 'C', 1, 2)
      sol8s = constant_renumber(sol8, 'C', 1, 2)
      sol9s = constant_renumber(sol9, 'C', 1, 3)
      sol10s = constant_renumber(sol10, 'C', 1, 2)
      sol11s = constant_renumber(sol11, 'C', 1, 2)
      sol12s = constant_renumber(sol12, 'C', 1, 4)
      assert dsolve(eq1, hint=hint) in (sol1, sol1s)
      assert dsolve(eq2, hint=hint) in (sol2, sol2s)
      assert dsolve(eq3, hint=hint) in (sol3, sol3s)
      assert dsolve(eq4, hint=hint) in (sol4, sol4s)
      assert dsolve(eq5, hint=hint) in (sol5, sol5s)
      assert dsolve(eq6, hint=hint) in (sol6, sol6s)
      assert dsolve(eq7, hint=hint) in (sol7, sol7s)
      assert dsolve(eq8, hint=hint) in (sol8, sol8s)
      assert dsolve(eq9, hint=hint) in (sol9, sol9s)
      assert dsolve(eq10, hint=hint) in (sol10, sol10s)
      assert dsolve(eq11, hint=hint+'_Integral') in (sol11, sol11s)
      assert dsolve(eq12, hint=hint) in (sol12, sol12s)
      assert checkodesol(eq1, sol1, order=3, solve_for_func=False)[0]
      assert checkodesol(eq2, sol2, order=3, solve_for_func=False)[0]
      assert checkodesol(eq3, sol3, order=1, solve_for_func=False)[0]
      assert checkodesol(eq4, sol4, order=2, solve_for_func=False)[0]
      assert checkodesol(eq5, sol5, order=2, solve_for_func=False)[0]
      assert checkodesol(eq6, sol6, order=2, solve_for_func=False)[0]
      assert checkodesol(eq7, sol7, order=2, solve_for_func=False)[0]
      assert checkodesol(eq8, sol8, order=2, solve_for_func=False)[0]
      assert checkodesol(eq9, sol9, order=3, solve_for_func=False)[0]
      assert checkodesol(eq10, sol10, order=2, solve_for_func=False)[0]
      assert checkodesol(eq12, sol12, order=4, solve_for_func=False)[0]
  
  def test_nth_linear_constant_coeff_variation_of_parameters_simplify_False():
      # solve_variation_of_parameters shouldn't attempt to simplify the
      # Wronskian if simplify=False.  If wronskian() ever gets good enough
      # to simplify the result itself, this test might fail.
      hint = 'nth_linear_constant_coeff_variation_of_parameters'
      assert dsolve(f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x) -
          2*x - exp(I*x), f(x), hint + "_Integral", simplify=False) != \
      dsolve(f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x) -
          2*x - exp(I*x), f(x), hint + "_Integral", simplify=True)
  
  def test_Liouville_ODE():
      hint = 'Liouville'
      # The first part here used to be test_ODE_1() from test_solvers.py
      eq1 = diff(f(x),x)/x + diff(f(x),x,x)/2 - diff(f(x),x)**2/2
      eq1a = diff(x*exp(-f(x)), x, x)
      # compare to test_unexpanded_Liouville_ODE() below
      eq2 = (eq1*exp(-f(x))/exp(f(x))).expand()
      eq3 = diff(f(x), x, x) + 1/f(x)*(diff(f(x), x))**2 + 1/x*diff(f(x), x)
      eq4 = x*diff(f(x), x, x) + x/f(x)*diff(f(x), x)**2 + x*diff(f(x), x)
      eq5 = Eq((x*exp(f(x))).diff(x, x), 0)
      sol1 = Eq(f(x), log(x/(C1 + C2*x)))
      sol1a = Eq(C1 + C2/x - exp(-f(x)), 0)
      sol2 = sol1
      sol3 = set([Eq(f(x), -sqrt(C1 + C2*log(x))), Eq(f(x), sqrt(C1 + C2*log(x)))])
      sol4 = set([Eq(f(x), sqrt(C1 + C2*exp(x))*exp(-x/2)),
                  Eq(f(x), -sqrt(C1 + C2*exp(x))*exp(-x/2))])
      sol5 = Eq(f(x), log(C1 + C2/x))
      sol1s = constant_renumber(sol1, 'C', 1, 2)
      sol2s = constant_renumber(sol2, 'C', 1, 2)
      sol3s = constant_renumber(sol3, 'C', 1, 2)
      sol4s = constant_renumber(sol4, 'C', 1, 2)
      sol5s = constant_renumber(sol5, 'C', 1, 2)
      assert dsolve(eq1, hint=hint) in (sol1, sol1s)
      assert dsolve(eq1a, hint=hint) in (sol1, sol1s)
      assert dsolve(eq2, hint=hint) in (sol2, sol2s)
      assert set(dsolve(eq3, hint=hint)) in (sol3, sol3s)
      assert set(dsolve(eq4, hint=hint)) in (sol4, sol4s)
      assert dsolve(eq5, hint=hint) in (sol5, sol5s)
      assert checkodesol(eq1, sol1, order=2, solve_for_func=False)[0]
      assert checkodesol(eq1a, sol1a, order=2, solve_for_func=False)[0]
      assert checkodesol(eq2, sol2, order=2, solve_for_func=False)[0]
      assert all(i[0] for i in checkodesol(eq3, sol3, order=2,
          solve_for_func=False))
      assert all(i[0] for i in checkodesol(eq4, sol4, order=2,
          solve_for_func=False))
      assert checkodesol(eq5, sol5, order=2, solve_for_func=False)[0]
      not_Liouville1 = classify_ode(diff(f(x),x)/x + f(x)*diff(f(x),x,x)/2 -
          diff(f(x),x)**2/2, f(x))
      not_Liouville2 = classify_ode(diff(f(x),x)/x + diff(f(x),x,x)/2 -
          x*diff(f(x),x)**2/2, f(x))
      assert hint not in not_Liouville1
      assert hint not in not_Liouville2
      assert hint+'_Integral' not in not_Liouville1
      assert hint+'_Integral' not in not_Liouville2
  
  def test_unexpanded_Liouville_ODE():
      # This is the same as eq1 from test_Liouville_ODE() above.
      eq1 = diff(f(x),x)/x+diff(f(x),x,x)/2- diff(f(x),x)**2/2
      eq2 = eq1*exp(-f(x))/exp(f(x))
      sol2 = Eq(f(x), log(x/(C1 + C2*x)))
      sol2s = constant_renumber(sol2, 'C', 1, 2)
      assert dsolve(eq2) in (sol2, sol2s)
      assert checkodesol(eq2, sol2, order=2, solve_for_func=False)[0]
  
  def test_1686():
      from sympy.abc import A
      eq = x + A*(x + diff(f(x), x) + f(x)) + diff(f(x), x) + f(x) + 2
      assert classify_ode(eq, f(x)) == ('1st_linear',
      'nth_linear_constant_coeff_undetermined_coefficients',
      'nth_linear_constant_coeff_variation_of_parameters',
      '1st_linear_Integral',
      'nth_linear_constant_coeff_variation_of_parameters_Integral')
      # 1765
      eq=(x**2 + f(x)**2)*f(x).diff(x) - 2*x*f(x)
      assert classify_ode(eq, f(x)) == (
          '1st_homogeneous_coeff_best',
          '1st_homogeneous_coeff_subs_indep_div_dep',
          '1st_homogeneous_coeff_subs_dep_div_indep',
          '1st_homogeneous_coeff_subs_indep_div_dep_Integral',
          '1st_homogeneous_coeff_subs_dep_div_indep_Integral')
  
  def test_1726():
      raises(ValueError, "dsolve(f(x, y).diff(x) - y*f(x, y), f(x))")
      assert classify_ode(f(x, y).diff(x) - y*f(x, y), f(x), dict=True) == \
      {'default': None, 'order': 0}
      # See also issue 694, test Z13.
      raises(ValueError, "dsolve(f(x).diff(x), f(y))")
      assert classify_ode(f(x).diff(x), f(y), dict=True) == \
          {'default': None, 'order': 0}
  
  def test_constant_renumber_order_issue2209():
      from sympy.utilities.iterables import variations
  
      assert constant_renumber(C1*x + C2*y, "C", 1, 2) == \
             constant_renumber(C1*y + C2*x, "C", 1, 2) == \
             C1*x + C2*y
      e = C1*(C2 + x)*(C3 + y)
      for a, b, c in variations([C1, C2, C3], 3):
          assert constant_renumber(a*(b + x)*(c + y), "C", 1, 3) == e
  
  def test_issue_2671():
      k = Symbol("k", real=True)
      t = Symbol('t')
      w = Function('w')
      sol = dsolve(w(t).diff(t, 6) - k**6*w(t), w(t))
      assert len([s for s in sol.atoms(Symbol) if s.name.startswith('C')]) == 6
      assert constantsimp((C1*cos(x) + C2*cos(x))*exp(x), x, 2) == \
          C1*cos(x)*exp(x)
      assert constantsimp(C1*cos(x) + C2*cos(x) + C3*sin(x), x, 2) == \
          C1*cos(x) + C3*sin(x)
      assert constantsimp(exp(C1 + x), x, 1) == C1*exp(x)
      assert constantsimp(2**(C1 + x), x, 1) == C1*2**x
      assert constantsimp(2**(C1 + x), x, 1) == C1*2**x
      assert constantsimp(x + C1 + y, x, 1) == C1 + x
      assert constantsimp(x + C1 + Integral(x, (x, 1, 2)), x, 1) == C1 + x
  
  def test_issue_2013_2331():
      assert homogeneous_order(-log(x) + acosh(x), x) is None
      assert homogeneous_order(y - log(x), x, y) is None
  