code: def dsolve(
      eq,
      func=None,
      hint="default",
      simplify=True,
      ics=None,
      xi=None,
      eta=None,
      x0=0,
      n=6,
      **kwargs,
  ):
      r"""
      Solves any (supported) kind of ordinary differential equation and
      system of ordinary differential equations.
  
      For single ordinary differential equation
      =========================================
  
      It is classified under this when number of equation in ``eq`` is one.
      **Usage**
  
          ``dsolve(eq, f(x), hint)`` -> Solve ordinary differential equation
          ``eq`` for function ``f(x)``, using method ``hint``.
  
      **Details**
  
          ``eq`` can be any supported ordinary differential equation (see the
              :py:mod:`~sympy.solvers.ode` docstring for supported methods).
              This can either be an :py:class:`~sympy.core.relational.Equality`,
              or an expression, which is assumed to be equal to ``0``.
  
          ``f(x)`` is a function of one variable whose derivatives in that
              variable make up the ordinary differential equation ``eq``.  In
              many cases it is not necessary to provide this; it will be
              autodetected (and an error raised if it could not be detected).
  
          ``hint`` is the solving method that you want dsolve to use.  Use
              ``classify_ode(eq, f(x))`` to get all of the possible hints for an
              ODE.  The default hint, ``default``, will use whatever hint is
              returned first by :py:meth:`~sympy.solvers.ode.classify_ode`.  See
              Hints below for more options that you can use for hint.
  
          ``simplify`` enables simplification by
              :py:meth:`~sympy.solvers.ode.ode.odesimp`.  See its docstring for more
              information.  Turn this off, for example, to disable solving of
              solutions for ``func`` or simplification of arbitrary constants.
              It will still integrate with this hint. Note that the solution may
              contain more arbitrary constants than the order of the ODE with
              this option enabled.
  
          ``xi`` and ``eta`` are the infinitesimal functions of an ordinary
              differential equation. They are the infinitesimals of the Lie group
              of point transformations for which the differential equation is
              invariant. The user can specify values for the infinitesimals. If
              nothing is specified, ``xi`` and ``eta`` are calculated using
              :py:meth:`~sympy.solvers.ode.infinitesimals` with the help of various
              heuristics.
  
          ``ics`` is the set of initial/boundary conditions for the differential equation.
            It should be given in the form of ``{f(x0): x1, f(x).diff(x).subs(x, x2):
            x3}`` and so on.  For power series solutions, if no initial
            conditions are specified ``f(0)`` is assumed to be ``C0`` and the power
            series solution is calculated about 0.
  
          ``x0`` is the point about which the power series solution of a differential
            equation is to be evaluated.
  
          ``n`` gives the exponent of the dependent variable up to which the power series
            solution of a differential equation is to be evaluated.
  
      **Hints**
  
          Aside from the various solving methods, there are also some meta-hints
          that you can pass to :py:meth:`~sympy.solvers.ode.dsolve`:
  
          ``default``:
                  This uses whatever hint is returned first by
                  :py:meth:`~sympy.solvers.ode.classify_ode`. This is the
                  default argument to :py:meth:`~sympy.solvers.ode.dsolve`.
  
          ``all``:
                  To make :py:meth:`~sympy.solvers.ode.dsolve` apply all
                  relevant classification hints, use ``dsolve(ODE, func,
                  hint="all")``.  This will return a dictionary of
                  ``hint:solution`` terms.  If a hint causes dsolve to raise the
                  ``NotImplementedError``, value of that hint's key will be the
                  exception object raised.  The dictionary will also include
                  some special keys:
  
                  - ``order``: The order of the ODE.  See also
                    :py:meth:`~sympy.solvers.deutils.ode_order` in
                    ``deutils.py``.
                  - ``best``: The simplest hint; what would be returned by
                    ``best`` below.
                  - ``best_hint``: The hint that would produce the solution
                    given by ``best``.  If more than one hint produces the best
                    solution, the first one in the tuple returned by
                    :py:meth:`~sympy.solvers.ode.classify_ode` is chosen.
                  - ``default``: The solution that would be returned by default.
                    This is the one produced by the hint that appears first in
                    the tuple returned by
                    :py:meth:`~sympy.solvers.ode.classify_ode`.
  
          ``all_Integral``:
                  This is the same as ``all``, except if a hint also has a
                  corresponding ``_Integral`` hint, it only returns the
                  ``_Integral`` hint.  This is useful if ``all`` causes
                  :py:meth:`~sympy.solvers.ode.dsolve` to hang because of a
                  difficult or impossible integral.  This meta-hint will also be
                  much faster than ``all``, because
                  :py:meth:`~sympy.core.expr.Expr.integrate` is an expensive
                  routine.
  
          ``best``:
                  To have :py:meth:`~sympy.solvers.ode.dsolve` try all methods
                  and return the simplest one.  This takes into account whether
                  the solution is solvable in the function, whether it contains
                  any Integral classes (i.e.  unevaluatable integrals), and
                  which one is the shortest in size.
  
          See also the :py:meth:`~sympy.solvers.ode.classify_ode` docstring for
          more info on hints, and the :py:mod:`~sympy.solvers.ode` docstring for
          a list of all supported hints.
  
      **Tips**
  
          - You can declare the derivative of an unknown function this way:
  
              >>> from sympy import Function, Derivative
              >>> from sympy.abc import x # x is the independent variable
              >>> f = Function("f")(x) # f is a function of x
              >>> # f_ will be the derivative of f with respect to x
              >>> f_ = Derivative(f, x)
  
          - See ``test_ode.py`` for many tests, which serves also as a set of
            examples for how to use :py:meth:`~sympy.solvers.ode.dsolve`.
          - :py:meth:`~sympy.solvers.ode.dsolve` always returns an
            :py:class:`~sympy.core.relational.Equality` class (except for the
            case when the hint is ``all`` or ``all_Integral``).  If possible, it
            solves the solution explicitly for the function being solved for.
            Otherwise, it returns an implicit solution.
          - Arbitrary constants are symbols named ``C1``, ``C2``, and so on.
          - Because all solutions should be mathematically equivalent, some
            hints may return the exact same result for an ODE. Often, though,
            two different hints will return the same solution formatted
            differently.  The two should be equivalent. Also note that sometimes
            the values of the arbitrary constants in two different solutions may
            not be the same, because one constant may have "absorbed" other
            constants into it.
          - Do ``help(ode.ode_<hintname>)`` to get help more information on a
            specific hint, where ``<hintname>`` is the name of a hint without
            ``_Integral``.
  
      For system of ordinary differential equations
      =============================================
  
      **Usage**
          ``dsolve(eq, func)`` -> Solve a system of ordinary differential
          equations ``eq`` for ``func`` being list of functions including
          `x(t)`, `y(t)`, `z(t)` where number of functions in the list depends
          upon the number of equations provided in ``eq``.
  
      **Details**
  
          ``eq`` can be any supported system of ordinary differential equations
          This can either be an :py:class:`~sympy.core.relational.Equality`,
          or an expression, which is assumed to be equal to ``0``.
  
          ``func`` holds ``x(t)`` and ``y(t)`` being functions of one variable which
          together with some of their derivatives make up the system of ordinary
          differential equation ``eq``. It is not necessary to provide this; it
          will be autodetected (and an error raised if it could not be detected).
  
      **Hints**
  
          The hints are formed by parameters returned by classify_sysode, combining
          them give hints name used later for forming method name.
  
      Examples
      ========
  
      >>> from sympy import Function, dsolve, Eq, Derivative, sin, cos, symbols
      >>> from sympy.abc import x
      >>> f = Function('f')
      >>> dsolve(Derivative(f(x), x, x) + 9*f(x), f(x))
      Eq(f(x), C1*sin(3*x) + C2*cos(3*x))
  
      >>> eq = sin(x)*cos(f(x)) + cos(x)*sin(f(x))*f(x).diff(x)
      >>> dsolve(eq, hint='1st_exact')
      [Eq(f(x), -acos(C1/cos(x)) + 2*pi), Eq(f(x), acos(C1/cos(x)))]
      >>> dsolve(eq, hint='almost_linear')
      [Eq(f(x), -acos(C1/cos(x)) + 2*pi), Eq(f(x), acos(C1/cos(x)))]
      >>> t = symbols('t')
      >>> x, y = symbols('x, y', cls=Function)
      >>> eq = (Eq(Derivative(x(t),t), 12*t*x(t) + 8*y(t)), Eq(Derivative(y(t),t), 21*x(t) + 7*t*y(t)))
      >>> dsolve(eq)
      [Eq(x(t), C1*x0(t) + C2*x0(t)*Integral(8*exp(Integral(7*t, t))*exp(Integral(12*t, t))/x0(t)**2, t)),
      Eq(y(t), C1*y0(t) + C2*(y0(t)*Integral(8*exp(Integral(7*t, t))*exp(Integral(12*t, t))/x0(t)**2, t) +
      exp(Integral(7*t, t))*exp(Integral(12*t, t))/x0(t)))]
      >>> eq = (Eq(Derivative(x(t),t),x(t)*y(t)*sin(t)), Eq(Derivative(y(t),t),y(t)**2*sin(t)))
      >>> dsolve(eq)
      {Eq(x(t), -exp(C1)/(C2*exp(C1) - cos(t))), Eq(y(t), -1/(C1 - cos(t)))}
      """
      if iterable(eq):
          from sympy.solvers.ode.systems import dsolve_system
  
          # This may have to be changed in future
          # when we have weakly and strongly
          # connected components. This have to
          # changed to show the systems that haven't
          # been solved.
          try:
              sol = dsolve_system(eq, funcs=func, ics=ics, doit=True)
              return sol[0] if len(sol) == 1 else sol
          except NotImplementedError:
              pass
  
          match = classify_sysode(eq, func)
  
          eq = match["eq"]
          order = match["order"]
          func = match["func"]
          t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0]
  
          # keep highest order term coefficient positive
          for i in range(len(eq)):
              for func_ in func:
                  if isinstance(func_, list):
                      pass
                  else:
                      if (
                          eq[i]
                          .coeff(
                              diff(
                                  func[i], t, ode_order(eq[i], func[i])
                              )
                          )
                          .is_negative
                      ):
                          eq[i] = -eq[i]
          match["eq"] = eq
          if len(set(order.values())) != 1:
              raise ValueError(
                  "It solves only those systems of equations whose orders are equal"
              )
          match["order"] = list(order.values())[0]
  
          def recur_len(l):
              return sum(
                  recur_len(item) if isinstance(item, list) else 1
                  for item in l
              )
  
          if recur_len(func) != len(eq):
              raise ValueError(
                  "dsolve() and classify_sysode() work with "
                  "number of functions being equal to number of equations"
              )
          if match["type_of_equation"] is None:
              raise NotImplementedError
          else:
              if match["is_linear"] == True:
                  solvefunc = globals()[
                      "sysode_linear_%(no_of_equation)seq_order%(order)s"
                      % match
                  ]
              else:
                  solvefunc = globals()[
                      "sysode_nonlinear_%(no_of_equation)seq_order%(order)s"
                      % match
                  ]
              sols = solvefunc(match)
              if ics:
                  constants = (
                      Tuple(*sols).free_symbols
                      - Tuple(*eq).free_symbols
                  )
                  solved_constants = solve_ics(
                      sols, func, constants, ics
                  )
                  return [sol.subs(solved_constants) for sol in sols]
              return sols
      else:
          given_hint = hint  # hint given by the user
  
          # See the docstring of _desolve for more details.
          hints = _desolve(
              eq,
              func=func,
              hint=hint,
              simplify=True,
              xi=xi,
              eta=eta,
              type="ode",
              ics=ics,
              x0=x0,
              n=n,
              **kwargs,
          )
          eq = hints.pop("eq", eq)
          all_ = hints.pop("all", False)
          if all_:
              retdict = {}
              failed_hints = {}
              gethints = classify_ode(eq, dict=True, hint="all")
              orderedhints = gethints["ordered_hints"]
              for hint in hints:
                  try:
                      rv = _helper_simplify(
                          eq, hint, hints[hint], simplify
                      )
                  except NotImplementedError as detail:
                      failed_hints[hint] = detail
                  else:
                      retdict[hint] = rv
              func = hints[hint]["func"]
  
              retdict["best"] = min(
                  list(retdict.values()),
                  key=lambda x: ode_sol_simplicity(
                      x, func, trysolving=not simplify
                  ),
              )
              if given_hint == "best":
                  return retdict["best"]
              for i in orderedhints:
                  if retdict["best"] == retdict.get(i, None):
                      retdict["best_hint"] = i
                      break
              retdict["default"] = gethints["default"]
              retdict["order"] = gethints["order"]
              retdict.update(failed_hints)
              return retdict
  
          else:
              # The key 'hint' stores the hint needed to be solved for.
              hint = hints["hint"]
              return _helper_simplify(
                  eq, hint, hints, simplify, ics=ics
              )
  
imports: from sympy.core import Add, S, Mul, Pow, oo
  from sympy.core.containers import Tuple
  from sympy.core.expr import AtomicExpr, Expr
  from sympy.core.function import Function, Derivative, AppliedUndef, diff, expand, expand_mul, Subs
  from sympy.core.multidimensional import vectorize
  from sympy.core.numbers import nan, zoo, Number
  from sympy.core.relational import Equality, Eq
  from sympy.core.sorting import default_sort_key, ordered
  from sympy.core.symbol import Symbol, Wild, Dummy, symbols
  from sympy.core.sympify import sympify
  from sympy.core.traversal import preorder_traversal
  from sympy.logic.boolalg import BooleanAtom, BooleanTrue, BooleanFalse
  from sympy.functions import exp, log, sqrt
  from sympy.functions.combinatorial.factorials import factorial
  from sympy.integrals.integrals import Integral
  from sympy.polys import Poly, terms_gcd, PolynomialError, lcm
  from sympy.polys.polytools import cancel
  from sympy.series import Order
  from sympy.series.series import series
  from sympy.simplify import collect, logcombine, powsimp, separatevars, simplify, cse
  from sympy.simplify.radsimp import collect_const
  from sympy.solvers import checksol, solve
  from sympy.utilities import numbered_symbols
  from sympy.utilities.iterables import uniq, sift, iterable
  from sympy.solvers.deutils import _preprocess, ode_order, _desolve
  from .single import SingleODEProblem, SingleODESolver, solver_map
label_desc: Use image node_id 3 for calling a global function with example usage: dsolve(eq, func, hint, simplify, ics, xi, eta, x0, n) and returns: unknown, sum, sols, retdict, _helper_simplify, unknown, retdict