Fundamental set of solutions for ODE

Question

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.

Answer 1

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 $a=1$.

Answer 2

If $0<a<b$, then $b\ge 2$ and thus $y(x)=x^b$ has values $y(0)=y'(0)=0$. However the only solution of an initial value problem for the given DE form with these initial conditions is the zero solution. As $x^b\ne 0$ in general, this gives a contradiction.