code: def test_pde_1st_linear_constant_coeff_homogeneous():
      f, F = map(Function, ["f", "F"])
      u = f(x, y)
      eq = 2 * u + u.diff(x) + u.diff(y)
      assert classify_pde(eq) == (
          "1st_linear_constant_coeff_homogeneous",
      )
      sol = pdsolve(eq)
      assert sol == Eq(u, F(x - y) * exp(-x - y))
      assert checkpdesol(eq, sol)[0]
  
      eq = 4 + (3 * u.diff(x) / u) + (2 * u.diff(y) / u)
      assert classify_pde(eq) == (
          "1st_linear_constant_coeff_homogeneous",
      )
      sol = pdsolve(eq)
      assert sol == Eq(
          u, F(2 * x - 3 * y) * exp(-S(12) * x / 13 - S(8) * y / 13)
      )
      assert checkpdesol(eq, sol)[0]
  
      eq = u + (6 * u.diff(x)) + (7 * u.diff(y))
      assert classify_pde(eq) == (
          "1st_linear_constant_coeff_homogeneous",
      )
      sol = pdsolve(eq)
      assert sol == Eq(
          u, F(7 * x - 6 * y) * exp(-6 * x / S(85) - 7 * y / S(85))
      )
      assert checkpdesol(eq, sol)[0]
  
      eq = a * u + b * u.diff(x) + c * u.diff(y)
      sol = pdsolve(eq)
      assert checkpdesol(eq, sol)[0]
  
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 8 for calling a global function with example usage: test_pde_1st_linear_constant_coeff_homogeneous() without return types