solving differential equation: (1+x^2)y``+2xy`=0 by substitution HELP!?

Question

Hi,

I need help solving (1+x^2)y``+2xy`=0 by substitution for the initial conditions y(0)=0 and y`(0) = 2.

I don't know that the problem statement means by substitution. Any ideas?

Answer 1

EDIT: Oops, made it much harder than it needs to be...the substitution is alright, but it results in not just a first order linear, but in the much easier separable case! If you still have trouble, feel free to e-mail me.

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Original:
The substitution is to lessen the degree of your equation. If you let w=y', then w'=y'', and suddenly you have a first order linear DE. After you've finished solving this for w, you simply integrate to get back to y. Don't forget to add your constant, although it won't matter here, as the initial conditions rid you of one of the constants. Your final answer should be

y=2*arctan(x) .

Double checking this answer verifies that it fills all of your conditions.