code: def test_classify_sysode():
      # Here x is assumed to be x(t) and y as y(t) for simplicity.
      # Similarly diff(x,t) and diff(y,y) is assumed to be x1 and y1 respectively.
      k, l, m, n = symbols("k, l, m, n", Integer=True)
      k1, k2, k3, l1, l2, l3, m1, m2, m3 = symbols(
          "k1, k2, k3, l1, l2, l3, m1, m2, m3", Integer=True
      )
      P, Q, R, p, q, r = symbols("P, Q, R, p, q, r", cls=Function)
      P1, P2, P3, Q1, Q2, R1, R2 = symbols(
          "P1, P2, P3, Q1, Q2, R1, R2", cls=Function
      )
      x, y, z = symbols("x, y, z", cls=Function)
      t = symbols("t")
      x1 = diff(x(t), t)
      y1 = diff(y(t), t)
  
      eq6 = (
          Eq(x1, exp(k * x(t)) * P(x(t), y(t))),
          Eq(y1, r(y(t)) * P(x(t), y(t))),
      )
      sol6 = {
          "no_of_equation": 2,
          "func_coeff": {
              (0, x(t), 0): 0,
              (1, x(t), 1): 0,
              (0, x(t), 1): 1,
              (1, y(t), 0): 0,
              (1, x(t), 0): 0,
              (0, y(t), 1): 0,
              (0, y(t), 0): 0,
              (1, y(t), 1): 1,
          },
          "type_of_equation": "type2",
          "func": [x(t), y(t)],
          "is_linear": False,
          "eq": [
              -P(x(t), y(t)) * exp(k * x(t)) + Derivative(x(t), t),
              -P(x(t), y(t)) * r(y(t)) + Derivative(y(t), t),
          ],
          "order": {y(t): 1, x(t): 1},
      }
      assert classify_sysode(eq6) == sol6
  
      eq7 = (Eq(x1, x(t) ** 2 + y(t) / x(t)), Eq(y1, x(t) / y(t)))
      sol7 = {
          "no_of_equation": 2,
          "func_coeff": {
              (0, x(t), 0): 0,
              (1, x(t), 1): 0,
              (0, x(t), 1): 1,
              (1, y(t), 0): 0,
              (1, x(t), 0): -1 / y(t),
              (0, y(t), 1): 0,
              (0, y(t), 0): -1 / x(t),
              (1, y(t), 1): 1,
          },
          "type_of_equation": "type3",
          "func": [x(t), y(t)],
          "is_linear": False,
          "eq": [
              -x(t) ** 2 + Derivative(x(t), t) - y(t) / x(t),
              -x(t) / y(t) + Derivative(y(t), t),
          ],
          "order": {y(t): 1, x(t): 1},
      }
      assert classify_sysode(eq7) == sol7
  
      eq8 = (
          Eq(x1, P1(x(t)) * Q1(y(t)) * R(x(t), y(t), t)),
          Eq(y1, P1(x(t)) * Q1(y(t)) * R(x(t), y(t), t)),
      )
      sol8 = {
          "func": [x(t), y(t)],
          "is_linear": False,
          "type_of_equation": "type4",
          "eq": [
              -P1(x(t)) * Q1(y(t)) * R(x(t), y(t), t)
              + Derivative(x(t), t),
              -P1(x(t)) * Q1(y(t)) * R(x(t), y(t), t)
              + Derivative(y(t), t),
          ],
          "func_coeff": {
              (0, y(t), 1): 0,
              (1, y(t), 1): 1,
              (1, x(t), 1): 0,
              (0, y(t), 0): 0,
              (1, x(t), 0): 0,
              (0, x(t), 0): 0,
              (1, y(t), 0): 0,
              (0, x(t), 1): 1,
          },
          "order": {y(t): 1, x(t): 1},
          "no_of_equation": 2,
      }
      assert classify_sysode(eq8) == sol8
  
      eq11 = (Eq(x1, x(t) * y(t) ** 3), Eq(y1, y(t) ** 5))
      sol11 = {
          "no_of_equation": 2,
          "func_coeff": {
              (0, x(t), 0): -y(t) ** 3,
              (1, x(t), 1): 0,
              (0, x(t), 1): 1,
              (1, y(t), 0): 0,
              (1, x(t), 0): 0,
              (0, y(t), 1): 0,
              (0, y(t), 0): 0,
              (1, y(t), 1): 1,
          },
          "type_of_equation": "type1",
          "func": [x(t), y(t)],
          "is_linear": False,
          "eq": [
              -x(t) * y(t) ** 3 + Derivative(x(t), t),
              -y(t) ** 5 + Derivative(y(t), t),
          ],
          "order": {y(t): 1, x(t): 1},
      }
      assert classify_sysode(eq11) == sol11
  
      eq13 = (
          Eq(x1, x(t) * y(t) * sin(t) ** 2),
          Eq(y1, y(t) ** 2 * sin(t) ** 2),
      )
      sol13 = {
          "no_of_equation": 2,
          "func_coeff": {
              (0, x(t), 0): -y(t) * sin(t) ** 2,
              (1, x(t), 1): 0,
              (0, x(t), 1): 1,
              (1, y(t), 0): 0,
              (1, x(t), 0): 0,
              (0, y(t), 1): 0,
              (0, y(t), 0): -x(t) * sin(t) ** 2,
              (1, y(t), 1): 1,
          },
          "type_of_equation": "type4",
          "func": [x(t), y(t)],
          "is_linear": False,
          "eq": [
              -x(t) * y(t) * sin(t) ** 2 + Derivative(x(t), t),
              -y(t) ** 2 * sin(t) ** 2 + Derivative(y(t), t),
          ],
          "order": {y(t): 1, x(t): 1},
      }
      assert classify_sysode(eq13) == sol13
  
imports: from sympy.core.function import Derivative, Function, Subs, diff
  from sympy.core.numbers import E, I, Rational, pi
  from sympy.core.relational import Eq
  from sympy.core.singleton import S
  from sympy.core.symbol import Symbol, symbols
  from sympy.functions.elementary.complexes import im, re
  from sympy.functions.elementary.exponential import exp, log
  from sympy.functions.elementary.hyperbolic import acosh
  from sympy.functions.elementary.miscellaneous import sqrt
  from sympy.functions.elementary.trigonometric import atan2, cos, sin, tan
  from sympy.integrals.integrals import Integral
  from sympy.polys.polytools import Poly
  from sympy.series.order import O
  from sympy.simplify.radsimp import collect
  from sympy.solvers.ode import classify_ode, homogeneous_order, dsolve
  from sympy.solvers.ode.subscheck import checkodesol
  from sympy.solvers.ode.ode import classify_sysode, constant_renumber, constantsimp, get_numbered_constants, solve_ics
  from sympy.solvers.ode.nonhomogeneous import _undetermined_coefficients_match
  from sympy.solvers.ode.single import LinearCoefficients
  from sympy.solvers.deutils import ode_order
  from sympy.testing.pytest import XFAIL, raises, slow, SKIP
  from sympy.utilities.misc import filldedent
label_desc: Use image node_id 7 for calling a global function with example usage: test_classify_sysode() without return types