Does there continuous functions $p(t),g(t)$ on $(-1,1)$ fullfill $y'' + p(x)y' +q(x)y = 0$ for $y(x) = c_1\cdot x^2 + c_2 \cdot x$?

Question

Does there continuous functions $p(x),g(x)$ on $(-1,1)$ fulfill $y'' + p(x)y' +q(x)y = 0$ for $y(x) = c_1\cdot x^2 + c_2 \cdot x$ ?

It's easy enough to show that $\{x^2,x\}$ are linearly independent on the interval $(-1,1)$, either directly showing or using a Wronksian.

But apart from doing that, i am not quite sure how to approach that question - yes, the functions are linearly independent - shall i just apply $y'',y',y $ for $y(x) = c_1 \cdot x^2+c_2 \cdot x$ to the equation and see what can i find ?

Thanks !

Answer 1

The answer is no.

We can solve for $p(x)$ and $q(x)$ knowing that $y=c_1x^2+ c_2 x$ is a solution.

We have $y=x$ and $y=x^2$ as solutions.

Upon substituting in our equation, we get $$2+2xp(x)+x^2q(x)=0$$ and $$ 0+p(x) + xq(x) =0$$

Solve for $p(x)$ and $q(x)$ we get, $p(x)=\frac {-2}{x}$ and $q(x) =\frac { 2}{x^2}$ none of which is continuous on (-1,1).