code: def test_nonlinear_2eq_order1():
      x, y, z = symbols("x, y, z", cls=Function)
      t = Symbol("t")
      eq1 = (
          Eq(diff(x(t), t), x(t) * y(t) ** 3),
          Eq(diff(y(t), t), y(t) ** 5),
      )
      sol1 = [
          Eq(
              x(t),
              C1 * exp((-1 / (4 * C2 + 4 * t)) ** (Rational(-1, 4))),
          ),
          Eq(y(t), -((-1 / (4 * C2 + 4 * t)) ** Rational(1, 4))),
          Eq(
              x(t),
              C1 * exp(-1 / (-1 / (4 * C2 + 4 * t)) ** Rational(1, 4)),
          ),
          Eq(y(t), (-1 / (4 * C2 + 4 * t)) ** Rational(1, 4)),
          Eq(
              x(t),
              C1 * exp(-I / (-1 / (4 * C2 + 4 * t)) ** Rational(1, 4)),
          ),
          Eq(y(t), -I * (-1 / (4 * C2 + 4 * t)) ** Rational(1, 4)),
          Eq(
              x(t),
              C1 * exp(I / (-1 / (4 * C2 + 4 * t)) ** Rational(1, 4)),
          ),
          Eq(y(t), I * (-1 / (4 * C2 + 4 * t)) ** Rational(1, 4)),
      ]
      assert dsolve(eq1) == sol1
      assert checksysodesol(eq1, sol1) == (True, [0, 0])
  
      eq2 = (
          Eq(diff(x(t), t), exp(3 * x(t)) * y(t) ** 3),
          Eq(diff(y(t), t), y(t) ** 5),
      )
      sol2 = [
          Eq(
              x(t),
              -log(C1 - 3 / (-1 / (4 * C2 + 4 * t)) ** Rational(1, 4))
              / 3,
          ),
          Eq(y(t), -((-1 / (4 * C2 + 4 * t)) ** Rational(1, 4))),
          Eq(
              x(t),
              -log(C1 + 3 / (-1 / (4 * C2 + 4 * t)) ** Rational(1, 4))
              / 3,
          ),
          Eq(y(t), (-1 / (4 * C2 + 4 * t)) ** Rational(1, 4)),
          Eq(
              x(t),
              -log(
                  C1 + 3 * I / (-1 / (4 * C2 + 4 * t)) ** Rational(1, 4)
              )
              / 3,
          ),
          Eq(y(t), -I * (-1 / (4 * C2 + 4 * t)) ** Rational(1, 4)),
          Eq(
              x(t),
              -log(
                  C1 - 3 * I / (-1 / (4 * C2 + 4 * t)) ** Rational(1, 4)
              )
              / 3,
          ),
          Eq(y(t), I * (-1 / (4 * C2 + 4 * t)) ** Rational(1, 4)),
      ]
      assert dsolve(eq2) == sol2
      assert checksysodesol(eq2, sol2) == (True, [0, 0])
  
      eq3 = (
          Eq(diff(x(t), t), y(t) * x(t)),
          Eq(diff(y(t), t), x(t) ** 3),
      )
      tt = Rational(2, 3)
      sol3 = [
          Eq(
              x(t),
              6**tt
              / (6 * (-sinh(sqrt(C1) * (C2 + t) / 2) / sqrt(C1)) ** tt),
          ),
          Eq(
              y(t),
              sqrt(C1 + C1 / sinh(sqrt(C1) * (C2 + t) / 2) ** 2) / 3,
          ),
      ]
      assert dsolve(eq3) == sol3
      # FIXME: assert checksysodesol(eq3, sol3) == (True, [0, 0])
  
      eq4 = (
          Eq(diff(x(t), t), x(t) * y(t) * sin(t) ** 2),
          Eq(diff(y(t), t), y(t) ** 2 * sin(t) ** 2),
      )
      sol4 = {
          Eq(x(t), -2 * exp(C1) / (C2 * exp(C1) + t - sin(2 * t) / 2)),
          Eq(y(t), -2 / (C1 + t - sin(2 * t) / 2)),
      }
      assert dsolve(eq4) == sol4
      # FIXME: assert checksysodesol(eq4, sol4) == (True, [0, 0])
  
      eq5 = (
          Eq(x(t), t * diff(x(t), t) + diff(x(t), t) * diff(y(t), t)),
          Eq(y(t), t * diff(y(t), t) + diff(y(t), t) ** 2),
      )
      sol5 = {Eq(x(t), C1 * C2 + C1 * t), Eq(y(t), C2**2 + C2 * t)}
      assert dsolve(eq5) == sol5
      assert checksysodesol(eq5, sol5) == (True, [0, 0])
  
      eq6 = (
          Eq(diff(x(t), t), x(t) ** 2 * y(t) ** 3),
          Eq(diff(y(t), t), y(t) ** 5),
      )
      sol6 = [
          Eq(
              x(t),
              1 / (C1 - 1 / (-1 / (4 * C2 + 4 * t)) ** Rational(1, 4)),
          ),
          Eq(y(t), -((-1 / (4 * C2 + 4 * t)) ** Rational(1, 4))),
          Eq(
              x(t),
              1 / (C1 + (-1 / (4 * C2 + 4 * t)) ** (Rational(-1, 4))),
          ),
          Eq(y(t), (-1 / (4 * C2 + 4 * t)) ** Rational(1, 4)),
          Eq(
              x(t),
              1 / (C1 + I / (-1 / (4 * C2 + 4 * t)) ** Rational(1, 4)),
          ),
          Eq(y(t), -I * (-1 / (4 * C2 + 4 * t)) ** Rational(1, 4)),
          Eq(
              x(t),
              1 / (C1 - I / (-1 / (4 * C2 + 4 * t)) ** Rational(1, 4)),
          ),
          Eq(y(t), I * (-1 / (4 * C2 + 4 * t)) ** Rational(1, 4)),
      ]
      assert dsolve(eq6) == sol6
      assert checksysodesol(eq6, sol6) == (True, [0, 0])
  
imports: from sympy.core.function import Derivative, Function, diff
  from sympy.core.mul import Mul
  from sympy.core.numbers import 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.hyperbolic import sinh
  from sympy.functions.elementary.miscellaneous import sqrt
  from sympy.matrices.dense import Matrix
  from sympy.core.containers import Tuple
  from sympy.functions import exp, cos, sin, log, Ci, Si, erf, erfi
  from sympy.matrices import dotprodsimp, NonSquareMatrixError
  from sympy.solvers.ode import dsolve
  from sympy.solvers.ode.ode import constant_renumber
  from sympy.solvers.ode.subscheck import checksysodesol
  from sympy.solvers.ode.systems import _classify_linear_system, linear_ode_to_matrix, ODEOrderError, ODENonlinearError, _simpsol, _is_commutative_anti_derivative, linodesolve, canonical_odes, dsolve_system, _component_division, _eqs2dict, _dict2graph
  from sympy.functions import airyai, airybi
  from sympy.integrals.integrals import Integral
  from sympy.simplify.ratsimp import ratsimp
  from sympy.testing.pytest import ON_CI, raises, slow, skip, XFAIL
label_desc: Use image node_id 41 for calling a global function with example usage: test_nonlinear_2eq_order1() without return types