# Buggy code
def _undetermined_coefficients_match(expr, x):
    r"""
    Returns a trial function match if undetermined coefficients can be applied
    to ``expr``, and ``None`` otherwise.

    A trial expression can be found for an expression for use with the method
    of undetermined coefficients if the expression is an
    additive/multiplicative combination of constants, polynomials in `x` (the
    independent variable of expr), `\sin(a x + b)`, `\cos(a x + b)`, and
    `e^{a x}` terms (in other words, it has a finite number of linearly
    independent derivatives).

    Note that you may still need to multiply each term returned here by
    sufficient `x` to make it linearly independent with the solutions to the
    homogeneous equation.

    This is intended for internal use by ``undetermined_coefficients`` hints.

    SymPy currently has no way to convert `\sin^n(x) \cos^m(y)` into a sum of
    only `\sin(a x)` and `\cos(b x)` terms, so these are not implemented.  So,
    for example, you will need to manually convert `\sin^2(x)` into `[1 +
    \cos(2 x)]/2` to properly apply the method of undetermined coefficients on
    it.

    Examples
    ========

    >>> from sympy import log, exp
    >>> from sympy.solvers.ode import _undetermined_coefficients_match
    >>> from sympy.abc import x
    >>> _undetermined_coefficients_match(9*x*exp(x) + exp(-x), x)
    {'test': True, 'trialset': {x*exp(x), exp(-x), exp(x)}}
    >>> _undetermined_coefficients_match(log(x), x)
    {'test': False}

    """
    a = Wild("a", exclude=[x])
    b = Wild("b", exclude=[x])
    expr = powsimp(expr, combine="exp")  # exp(x)*exp(2*x + 1) => exp(3*x + 1)
    retdict = {}

    def _test_term(expr, x):
        r"""
        Test if ``expr`` fits the proper form for undetermined coefficients.
        """
        if not expr.has(x):
            return True
        elif expr.is_Add:
            return all(_test_term(i, x) for i in expr.args)
        elif expr.is_Mul:
            if expr.has(sin, cos):
                foundtrig = False
                # Make sure that there is only one trig function in the args.
                # See the docstring.
                for i in expr.args:
                    if i.has(sin, cos):
                        if foundtrig:
                            return False
                        else:
                            foundtrig = True
            return all(_test_term(i, x) for i in expr.args)
        elif expr.is_Function:
            if expr.func in (sin, cos, exp):
                if expr.args[0].match(a * x + b):
                    return True
                else:
                    return False
            else:
                return False
        elif (
            expr.is_Pow
            and expr.base.is_Symbol
            and expr.exp.is_Integer
            and expr.exp >= 0
        ):
            return True
        elif expr.is_Pow and expr.base.is_number:
            if expr.exp.match(a * x + b):
                return True
            else:
                return False
        elif expr.is_Symbol or expr.is_number:
            return True
        else:
            return False

    def _get_trial_set(expr, x, exprs=set([])):
        r"""
        Returns a set of trial terms for undetermined coefficients.

        The idea behind undetermined coefficients is that the terms expression
        repeat themselves after a finite number of derivatives, except for the
        coefficients (they are linearly dependent).  So if we collect these,
        we should have the terms of our trial function.
        """

        def _remove_coefficient(expr, x):
            r"""
            Returns the expression without a coefficient.

            Similar to expr.as_independent(x)[1], except it only works
            multiplicatively.
            """
            term = S.One
            if expr.is_Mul:
                for i in expr.args:
                    if i.has(x):
                        term *= i
            elif expr.has(x):
                term = expr
            return term

        expr = expand_mul(expr)
        if expr.is_Add:
            for term in expr.args:
                if _remove_coefficient(term, x) in exprs:
                    pass
                else:
                    exprs.add(_remove_coefficient(term, x))
                    exprs = exprs.union(_get_trial_set(term, x, exprs))
        else:
            term = _remove_coefficient(expr, x)
            tmpset = exprs.union({term})
            oldset = set([])
            while tmpset != oldset:
                # If you get stuck in this loop, then _test_term is probably
                # broken
                oldset = tmpset.copy()
                expr = expr.diff(x)
                term = _remove_coefficient(expr, x)
                if term.is_Add:
                    tmpset = tmpset.union(_get_trial_set(term, x, tmpset))
                else:
                    tmpset.add(term)
            exprs = tmpset
        return exprs

    retdict["test"] = _test_term(expr, x)
    if retdict["test"]:
        # Try to generate a list of trial solutions that will have the
        # undetermined coefficients. Note that if any of these are not linearly
        # independent with any of the solutions to the homogeneous equation,
        # then they will need to be multiplied by sufficient x to make them so.
        # This function DOES NOT do that (it doesn't even look at the
        # homogeneous equation).
        retdict["trialset"] = _get_trial_set(expr, x)

    return retdict


# Traceback (NotImplementedError)
dsolve(f(x).diff(x, x) + f(x) - x*sin(x - 2), f(x), hint='nth_linear_constant_coeff_undetermined_coefficients')
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "sympy/solvers/ode.py", line 457, in dsolve
match=hints[hint]), func, hints['order'], hint)
File "sympy/solvers/ode.py", line 2370, in ode_nth_linear_constant_coeff_undetermined_coefficients
return _solve_undetermined_coefficients(eq, func, order, match)
File "sympy/solvers/ode.py", line 2460, in _solve_undetermined_coefficients
" method of undetermined coefficients (unable to solve for coefficients).")
NotImplementedError: Could not solve -x*sin(-2 + x) + D(f(x), x, x) + f(x) using the  method of undetermined coefficients (unable to solve for coefficients).

# Fixed code
def _undetermined_coefficients_match(expr, x, func=None, eq_homogeneous=S.Zero):
    r"""
    Returns a trial function match if undetermined coefficients can be applied
    to ``expr``, and ``None`` otherwise.

    A trial expression can be found for an expression for use with the method
    of undetermined coefficients if the expression is an
    additive/multiplicative combination of constants, polynomials in `x` (the
    independent variable of expr), `\sin(a x + b)`, `\cos(a x + b)`, and
    `e^{a x}` terms (in other words, it has a finite number of linearly
    independent derivatives).

    Note that you may still need to multiply each term returned here by
    sufficient `x` to make it linearly independent with the solutions to the
    homogeneous equation.

    This is intended for internal use by ``undetermined_coefficients`` hints.

    SymPy currently has no way to convert `\sin^n(x) \cos^m(y)` into a sum of
    only `\sin(a x)` and `\cos(b x)` terms, so these are not implemented.  So,
    for example, you will need to manually convert `\sin^2(x)` into `[1 +
    \cos(2 x)]/2` to properly apply the method of undetermined coefficients on
    it.

    Examples
    ========

    >>> from sympy import log, exp
    >>> from sympy.solvers.ode import _undetermined_coefficients_match
    >>> from sympy.abc import x
    >>> _undetermined_coefficients_match(9*x*exp(x) + exp(-x), x)
    {'test': True, 'trialset': {x*exp(x), exp(-x), exp(x)}}
    >>> _undetermined_coefficients_match(log(x), x)
    {'test': False}

    """
    a = Wild("a", exclude=[x])
    b = Wild("b", exclude=[x])
    expr = powsimp(expr, combine="exp")  # exp(x)*exp(2*x + 1) => exp(3*x + 1)
    retdict = {}

    def _test_term(expr, x):
        r"""
        Test if ``expr`` fits the proper form for undetermined coefficients.
        """
        if not expr.has(x):
            return True
        elif expr.is_Add:
            return all(_test_term(i, x) for i in expr.args)
        elif expr.is_Mul:
            if expr.has(sin, cos):
                foundtrig = False
                # Make sure that there is only one trig function in the args.
                # See the docstring.
                for i in expr.args:
                    if i.has(sin, cos):
                        if foundtrig:
                            return False
                        else:
                            foundtrig = True
            return all(_test_term(i, x) for i in expr.args)
        elif expr.is_Function:
            if expr.func in (sin, cos, exp):
                if expr.args[0].match(a * x + b):
                    return True
                else:
                    return False
            else:
                return False
        elif (
            expr.is_Pow
            and expr.base.is_Symbol
            and expr.exp.is_Integer
            and expr.exp >= 0
        ):
            return True
        elif expr.is_Pow and expr.base.is_number:
            if expr.exp.match(a * x + b):
                return True
            else:
                return False
        elif expr.is_Symbol or expr.is_number:
            return True
        else:
            return False

    def _get_trial_set(expr, x, exprs=set([])):
        r"""
        Returns a set of trial terms for undetermined coefficients.

        The idea behind undetermined coefficients is that the terms expression
        repeat themselves after a finite number of derivatives, except for the
        coefficients (they are linearly dependent).  So if we collect these,
        we should have the terms of our trial function.
        """

        def _remove_coefficient(expr, x):
            r"""
            Returns the expression without a coefficient.

            Similar to expr.as_independent(x)[1], except it only works
            multiplicatively.
            """
            term = S.One
            if expr.is_Mul:
                for i in expr.args:
                    if i.has(x):
                        term *= i
            elif expr.has(x):
                term = expr
            return term

        expr = expand_mul(expr)
        if expr.is_Add:
            for term in expr.args:
                if _remove_coefficient(term, x) in exprs:
                    pass
                else:
                    exprs.add(_remove_coefficient(term, x))
                    exprs = exprs.union(_get_trial_set(term, x, exprs))
        else:
            term = _remove_coefficient(expr, x)
            tmpset = exprs.union({term})
            oldset = set([])
            while tmpset != oldset:
                # If you get stuck in this loop, then _test_term is probably
                # broken
                oldset = tmpset.copy()
                expr = expr.diff(x)
                term = _remove_coefficient(expr, x)
                if term.is_Add:
                    tmpset = tmpset.union(_get_trial_set(term, x, tmpset))
                else:
                    tmpset.add(term)
            exprs = tmpset
        return exprs

    def is_homogeneous_solution(term):
        r"""This function checks whether the given trialset contains any root
        of homogenous equation"""
        return expand(sub_func_doit(eq_homogeneous, func, term)).is_zero

    retdict["test"] = _test_term(expr, x)
    if retdict["test"]:
        # Try to generate a list of trial solutions that will have the
        # undetermined coefficients. Note that if any of these are not linearly
        # independent with any of the solutions to the homogeneous equation,
        # then they will need to be multiplied by sufficient x to make them so.
        # This function DOES NOT do that (it doesn't even look at the
        # homogeneous equation).
        temp_set = set([])
        for i in Add.make_args(expr):
            act = _get_trial_set(i, x)
            if eq_homogeneous is not S.Zero:
                while any(is_homogeneous_solution(ts) for ts in act):
                    act = {x * ts for ts in act}
            temp_set = temp_set.union(act)

        retdict["trialset"] = temp_set

    return retdict