Finding a Second Solution to ODE xy''+y'+xy=0 with Integral Method

[Tangent87]

We're given the ODE xy''+y'+xy=0 and told that $$y=\int_0^{\pi} e^{ix\cos{t}}dt$$ is one solution and it asks to find a second solution in the form of an integral for x>0. I'm not sure how to do this, I don't think they mean the second solution derived from the Wronskian as that just wouldn't "look right" with an integral for the first solution? Also I've tried substituting in $$y=\int_{\gamma} f(t)e^{xt}dt$$ but I just get back to the solution they've already given us. Do you think you have to somehow just "spot" a second solution? Thanks.

 

[Gib Z]

You'll want to read this: http://en.wikipedia.org/wiki/Reduction_of_order

 

[Tangent87]

But that implies the second solution is:

$$y_2(x)=y_1(x)\int^x \frac{1}{uy_1(u)^2}du$$ where $$y_1(x)=\int_0^{\pi} e^{ix\cos{t}}dt$$. Is that okay? I've been a bit suspect when we have expressions where we're dividing by integrals.

 

[Gib Z]

It's fine. The integral is just some function.

 

[Tangent87]

It's fine. The integral is just some function.

Ok, thanks.