Second Order ODE Method Question

$x^2 \frac{d^2y}{dx^2}-2y=r(x)$

Part a) was to show $y=x^2$ is a solution for r(x)=0.

Part b) asks to use the method of variation of parameters to find the general solution when $r(x)=3x^2$.

In my notes, it says we use the method of variation of parameters when we know both solutions to the homogeneous equation (so both solutions when r(x)=0).

However, I don't know how you would find the other solution? I was going to use the method of reduction of order to find the second solution, but wouldn't this construct the general solution that I need to find?

TLDR: For the ODE above, given I know one solution for r(x)=0, should I use the method of reduction of order to find a solution for $r(x)=3x^2$?

Yes, I imagine that would work fine. (The other complementary function is also very simple.)

How would you work out the other cf?

$y_2 = u y_1 = u x^2$, and work from there, substituting it into the DE. (It came out for me in about five lines.)

But is this not the method of variation of parameters?

Thanks for the help anyway, I'm going to email my lecturer. I can't rep you, PRSOM. Thumbs up instead.

$x^2 \frac{d^2y}{dx^2}-2y=r(x)$

Part a) was to show $y=x^2$ is a solution for r(x)=0.

Part b) asks to use the method of variation of parameters to find the general solution when $r(x)=3x^2$.

In my notes, it says we use the method of variation of parameters when we know both solutions to the homogeneous equation (so both solutions when r(x)=0).

However, I don't know how you would find the other solution? I was going to use the method of reduction of order to find the second solution, but wouldn't this construct the general solution that I need to find?

TLDR: For the ODE above, given I know one solution for r(x)=0, should I use the method of reduction of order to find a solution for $r(x)=3x^2$?

Thanks no, variation of parameters involves knowing all the complementary functions and constructing a solution for a particular integral. I described a way of finding one complementary function if we already know only one.

I meant to say the method of reduction of order! (Such similar names...)