How can I solve this variable-coefficient ODE system?

Question

I originally have a linear, homogeneous, second-order variable coefficient ODE system of this form:

$X''(x) = A(x)X(x)$, where

$X(x) = $\begin{bmatrix} f(x) \\ g(x) \\ \end{bmatrix}

and

$A(x) = $\begin{bmatrix} \frac{\lambda+\alpha}{x(1-x)} & -\frac{\lambda}{x(1-x)} \\ -p^2 & p^2 \\ \end{bmatrix}

So far, I've tried to reduce this system to the normal form shown below, but without any particular success.

$X'(x) = A(x)X(x)$, where

$X(x) = $\begin{bmatrix} f_1(x) \\ f_2(x) \\ g_1(x) \\ g_2(x) \end{bmatrix}

and

$A(x) = $\begin{bmatrix} 0 & 1 & 0 & 0 \\ \frac{\lambda+\alpha}{x(1-x)} &0& -\frac{\lambda}{x(1-x)} & 0 \\ 0&0&0&1\\ -p^2 &0& p^2 & 0 \\ \end{bmatrix}

I know that this broadly lies under the theory of time-varying systems, but have little idea how to go about looking for exact analytical solutions.

Thank you!