Fundamental set of solutions for ODE Let $a$ and $b$ be distinct positive integers. Prove that $(x^{a}, x^{b})$ cannot be a fundemental set of solutions of any second order ODE of the form
$y''+p(x)y'+q(x)y=0$
on the interval $(-1,1)$, where $p(x)$ and $q(x)$ are continuous functions on $(-1,1)$.
My progress: I managed to obtain functions $p(x)=\frac{1-a-b}{x}$ and $q(x)=\frac{ab}{x^2}$, which are continuous everywhere except zero. Therefore, I am not sure how to continue after this, unsure whether this finishes the problem or not.

With $x^a$, $a>0$, we have
$$a(a-1)x^{a-2}+p(x)ax^{a-1}+q(x)x^a=0$$
and
$$\lim_{x\to 0}(a(a-1)+xp(x)+ax^2q(x))=a(a-1)=0,$$ which is only possible with $a=1$.
