I was given the following problem:
Let $y''+p(x)y'+q(x)y=r(x)$ be a second order differential equation, where $p(x),q(x),r(x)$ are continuous on $(-\pi,\pi)$ and suppose $u(x) = x^2\sin x$ is a solution to the equation. prove that the equation is nonhomogeneous
I tried to suppose by contradiction that $r(x)= 0$ and assume the existance of another solution, $u_2(x)$, linearly independent of $u(x)$, and show that their Wronskian has both a zero and a nonzero value in $(-\pi,\pi)$. I was able to get a zero value, at $x=0$, but couldn't find a non-zero value, because I do not know what $u_2(x)$ and $u_2'(x)$ are.