code: def test_pde_1st_linear_constant_coeff():
      f, F = map(Function, ["f", "F"])
      u = f(x, y)
      eq = -2 * u.diff(x) + 4 * u.diff(y) + 5 * u - exp(x + 3 * y)
      sol = pdsolve(eq)
      assert sol == Eq(
          f(x, y),
          (F(4 * x + 2 * y) * exp(x / 2) + exp(x + 4 * y) / 15)
          * exp(-y),
      )
      assert classify_pde(eq) == (
          "1st_linear_constant_coeff",
          "1st_linear_constant_coeff_Integral",
      )
      assert checkpdesol(eq, sol)[0]
  
      eq = (u.diff(x) / u) + (u.diff(y) / u) + 1 - (exp(x + y) / u)
      sol = pdsolve(eq)
      assert sol == Eq(
          f(x, y), F(x - y) * exp(-x / 2 - y / 2) + exp(x + y) / 3
      )
      assert classify_pde(eq) == (
          "1st_linear_constant_coeff",
          "1st_linear_constant_coeff_Integral",
      )
      assert checkpdesol(eq, sol)[0]
  
      eq = 2 * u + -u.diff(x) + 3 * u.diff(y) + sin(x)
      sol = pdsolve(eq)
      assert sol == Eq(
          f(x, y),
          F(3 * x + y) * exp(x / 5 - 3 * y / 5)
          - 2 * sin(x) / 5
          - cos(x) / 5,
      )
      assert classify_pde(eq) == (
          "1st_linear_constant_coeff",
          "1st_linear_constant_coeff_Integral",
      )
      assert checkpdesol(eq, sol)[0]
  
      eq = u + u.diff(x) + u.diff(y) + x * y
      sol = pdsolve(eq)
      assert (
          sol.expand()
          == Eq(
              f(x, y),
              x
              + y
              + (x - y) ** 2 / 4
              - (x + y) ** 2 / 4
              + F(x - y) * exp(-x / 2 - y / 2)
              - 2,
          ).expand()
      )
      assert classify_pde(eq) == (
          "1st_linear_constant_coeff",
          "1st_linear_constant_coeff_Integral",
      )
      assert checkpdesol(eq, sol)[0]
      eq = u + u.diff(x) + u.diff(y) + log(x)
      assert classify_pde(eq) == (
          "1st_linear_constant_coeff",
          "1st_linear_constant_coeff_Integral",
      )
  
file_path: sympy/sympy/solvers/tests/test_pde.py
imports: from sympy.core.function import Derivative, Function
  from sympy.core.relational import Eq
  from sympy.core.symbol import Symbol, symbols
  from sympy.functions.elementary.exponential import exp, log
  from sympy.functions.elementary.trigonometric import cos, sin
  from sympy.core import S
  from sympy.solvers.pde import pde_separate, pde_separate_add, pde_separate_mul, pdsolve, classify_pde, checkpdesol
  from sympy.testing.pytest import raises
label_desc: Use image node_id 9 for calling a global function with example usage: test_pde_1st_linear_constant_coeff() without return types