code: def linodesolve_type(A, t, b=None):
      r"""
      Helper function that determines the type of the system of ODEs for solving with :obj:`sympy.solvers.ode.systems.linodesolve()`
  
      Explanation
      ===========
  
      This function takes in the coefficient matrix and/or the non-homogeneous term
      and returns the type of the equation that can be solved by :obj:`sympy.solvers.ode.systems.linodesolve()`.
  
      If the system is constant coefficient homogeneous, then "type1" is returned
  
      If the system is constant coefficient non-homogeneous, then "type2" is returned
  
      If the system is non-constant coefficient homogeneous, then "type3" is returned
  
      If the system is non-constant coefficient non-homogeneous, then "type4" is returned
  
      If the system has a non-constant coefficient matrix which can be factorized into constant
      coefficient matrix, then "type5" or "type6" is returned for when the system is homogeneous or
      non-homogeneous respectively.
  
      Note that, if the system of ODEs is of "type3" or "type4", then along with the type,
      the commutative antiderivative of the coefficient matrix is also returned.
  
      If the system cannot be solved by :obj:`sympy.solvers.ode.systems.linodesolve()`, then
      NotImplementedError is raised.
  
      Parameters
      ==========
  
      A : Matrix
          Coefficient matrix of the system of ODEs
      b : Matrix or None
          Non-homogeneous term of the system. The default value is None.
          If this argument is None, then the system is assumed to be homogeneous.
  
      Examples
      ========
  
      >>> from sympy import symbols, Matrix
      >>> from sympy.solvers.ode.systems import linodesolve_type
      >>> t = symbols("t")
      >>> A = Matrix([[1, 1], [2, 3]])
      >>> b = Matrix([t, 1])
  
      >>> linodesolve_type(A, t)
      {'antiderivative': None, 'type_of_equation': 'type1'}
  
      >>> linodesolve_type(A, t, b=b)
      {'antiderivative': None, 'type_of_equation': 'type2'}
  
      >>> A_t = Matrix([[1, t], [-t, 1]])
  
      >>> linodesolve_type(A_t, t)
      {'antiderivative': Matrix([
      [      t, t**2/2],
      [-t**2/2,      t]]), 'type_of_equation': 'type3'}
  
      >>> linodesolve_type(A_t, t, b=b)
      {'antiderivative': Matrix([
      [      t, t**2/2],
      [-t**2/2,      t]]), 'type_of_equation': 'type4'}
  
      >>> A_non_commutative = Matrix([[1, t], [t, -1]])
      >>> linodesolve_type(A_non_commutative, t)
      Traceback (most recent call last):
      ...
      NotImplementedError:
      The system does not have a commutative antiderivative, it cannot be
      solved by linodesolve.
  
      Returns
      =======
  
      Dict
  
      Raises
      ======
  
      NotImplementedError
          When the coefficient matrix does not have a commutative antiderivative
  
      See Also
      ========
  
      linodesolve: Function for which linodesolve_type gets the information
  
      """
  
      match = {}
      is_non_constant = not _matrix_is_constant(A, t)
      is_non_homogeneous = not (b is None or b.is_zero_matrix)
      type = "type{}".format(
          int(
              "{}{}".format(
                  int(is_non_constant), int(is_non_homogeneous)
              ),
              2,
          )
          + 1
      )
  
      B = None
      match.update({"type_of_equation": type, "antiderivative": B})
  
      if is_non_constant:
          B, is_commuting = _is_commutative_anti_derivative(A, t)
          if not is_commuting:
              raise NotImplementedError(
                  filldedent(
                      """
                  The system does not have a commutative antiderivative, it cannot be solved
                  by linodesolve.
              """
                  )
              )
  
          match["antiderivative"] = B
          match.update(_first_order_type5_6_subs(A, t, b=b))
  
      return match
  
imports: from sympy.core import Add, Mul, S
  from sympy.core.containers import Tuple
  from sympy.core.exprtools import factor_terms
  from sympy.core.numbers import I
  from sympy.core.relational import Eq, Equality
  from sympy.core.sorting import default_sort_key, ordered
  from sympy.core.symbol import Dummy, Symbol
  from sympy.core.function import expand_mul, expand, Derivative, AppliedUndef, Function, Subs
  from sympy.functions import exp, im, cos, sin, re, Piecewise, piecewise_fold, sqrt, log
  from sympy.functions.combinatorial.factorials import factorial
  from sympy.matrices import zeros, Matrix, NonSquareMatrixError, MatrixBase, eye
  from sympy.polys import Poly, together
  from sympy.simplify import collect, radsimp, signsimp
  from sympy.simplify.powsimp import powdenest, powsimp
  from sympy.simplify.ratsimp import ratsimp
  from sympy.simplify.simplify import simplify
  from sympy.sets.sets import FiniteSet
  from sympy.solvers.deutils import ode_order
  from sympy.solvers.solveset import NonlinearError, solveset
  from sympy.utilities.iterables import connected_components, iterable, strongly_connected_components
  from sympy.utilities.misc import filldedent
  from sympy.integrals.integrals import Integral, integrate
label_desc: Use image node_id 5 for calling a global function with example usage: linodesolve_type(A, t, b) and returns: match