Determining the properties of solutions of a second order non homogenous ODE

Question

Consider the following second order homogeneous ODE: $$x''+p(t)x'+q(t)x=0$$

Prove that if $x_1(t),x_2(t)$ are two linearly independent solutions of the equation, then $p(t),q(t)$ depend only on $x_1(t),x_2(t)$

What i tried Since $x_1(t),x_2(t)$ are two linearly independent solutions, the workskain of $ y_1$ and $y_2$ must not equals to $0$. Hence $$w(x_1,x_2)(t)=det \left( \begin{smallmatrix} x_{1} &x_{2}\\ x'_{1} & x'_{2} \\\end{smallmatrix} \right)\neq0$$ which then gives which then gives $$x_1x'_2-x'_1x_2\neq0$$. Im stuck from here onwards,as im unsure of how to relate this result to $p(t),q(t)$ in order to be able to prove the above theorem. Could anyone please explain. Thanks

Answer 1

The differential equations for both solutions form a linear 2x2 system for p and q. And indeed the Wronskian is the determinant of that system.

I'm hopeful that you can solve 2x2 linear systems.