Repository: sympy/sympy
File: sympy/solvers/ode/single.py (Python)

Code before:
            return False

        # Store the matched values
        self.match_dict = {
            'A': pattern_match[A_sym],
            'k': pattern_match[k_sym],
        }

        return True

    def _get_general_solution(self, *, simplify_flag: bool = True):
        f = self.ode_problem.func.func
        x = self.ode_problem.sym
        (C1, C2) = self.ode_problem.get_numbered_constants(num=2)

        A = self.match_dict['A']
        k = self.match_dict['k']

        # Special case: k = -2 causes division by zero
        # The equation x^(-2)*y + y'' = 0 is an Euler equation
        # and should be handled differently
        if k == -2:
            # For x^(-2)*y + y'' = 0, characteristic equation is r^2 - r + A = 0
            discriminant = 1 - 4*A
            omega = sqrt(Abs(discriminant)) / 2
            return [Eq(f(x), sqrt(x)*(C1*sin(omega*log(x)) + C2*cos(omega*log(x))))]

        # For y'' + A*x^k*y = 0
        # The Bessel order and argument are:
        nu = 1 / (k + 2)

        z = (2 * sqrt(Abs(A)) / (k + 2)) * x**((k + 2)/2)

        # General solution
        return [Eq(f(x), sqrt(x) * (C1*besselj(nu, z) + C2*bessely(nu, z)))]


class SecondLinearAiry(SingleODESolver):
    r"""
    Gives solution of the Airy differential equation

    .. math :: \frac{d^2y}{dx^2} + (a + b x) y(x) = 0

    in terms of Airy special functions airyai and airybi.

    Examples
    ========

    >>> from sympy import dsolve, Function
    >>> from sympy.abc import x
    >>> f = Function("f")

Review comment (suggestion):
Is the `Abs` here needed and is it correct?

This gives
```python
In [10]: eq = A*x**(-2)*f(x) + f(x).diff(x, 2)

In [11]: dsolve(eq, hint='2nd_linear_bessel_symbolic')
Out[11]: 
          ⎛      ⎛         ___________⎞         ⎛         ___________⎞⎞
          ⎜      ⎜log(x)⋅╲╱ │4⋅A - 1│ ⎟         ⎜log(x)⋅╲╱ │4⋅A - 1│ ⎟⎟
f(x) = √x⋅⎜C₁⋅sin⎜────────────────────⎟ + C₂⋅cos⎜────────────────────⎟⎟
          ⎝      ⎝         2          ⎠         ⎝         2          ⎠⎠
```
For particular concrete values of `A` we can have
```python
In [15]: eq = x**(-2)*f(x)/2 + f(x).diff(x, 2)

In [16]: dsolve(eq)
Out[16]: 
          ⎛      ⎛log(x)⎞         ⎛log(x)⎞⎞
f(x) = √x⋅⎜C₁⋅sin⎜──────⎟ + C₂⋅cos⎜──────⎟⎟
          ⎝      ⎝  2   ⎠         ⎝  2   ⎠⎠

In [17]: eq = x**(-2)*f(x)/4 + f(x).diff(x, 2)

In [18]: dsolve(eq)
Out[18]: f(x) = √x⋅(C₁ + C₂⋅log(x))

In [19]: eq = x**(-2)*f(x)/8 + f(x).diff(x, 2)

In [20]: dsolve(eq)
Out[20]: 
           1   √2       √2   1
           ─ - ──       ── + ─
           2   4        4    2
f(x) = C₁⋅x       + C₂⋅x 
```
Maybe it is better just to reject this case when matching.

Code after:

        k_val = pattern_match[k_sym]
        A_val = pattern_match[A_sym]

        if k_val == -2:
            # Special case: k = -2 is an Euler equation, better handled by other solvers.
            return False

        # Store the matched values
        self.match_dict = {
            'A': A_val,
            'k': k_val,
        }

        return True

    def _get_general_solution(self, *, simplify_flag: bool = True):
        f = self.ode_problem.func.func
        x = self.ode_problem.sym
        (C1, C2) = self.ode_problem.get_numbered_constants(num=2)

        A = self.match_dict['A']
        k = self.match_dict['k']

        # For y'' + A*x^k*y = 0
        # The Bessel order and argument are:
        nu = 1 / (k + 2)

        z = (2 * sqrt(Abs(A)) / (k + 2)) * x**((k + 2)/2)

        # General solution
        return [Eq(f(x), sqrt(x) * (C1*besselj(nu, z) + C2*bessely(nu, z)))]


class SecondLinearAiry(SingleODESolver):
    r"""
    Gives solution of the Airy differential equation

    .. math :: \frac{d^2y}{dx^2} + (a + b x) y(x) = 0

    in terms of Airy special functions airyai and airybi.

    Examples
    ========

    >>> from sympy import dsolve, Function
    >>> from sympy.abc import x
    >>> f = Function("f")
    >>> eq = f(x).diff(x, 2) - x*f(x)
    >>> dsolve(eq)
    Eq(f(x), C1*airyai(x) + C2*airybi(x))