Find a particular solution of y"+4y=-12cos2x-4sin2x.

Question

idealist10 · Answer

@ganeshie8 @zepdrix @mathmale

kainui · Answer

There are several ways to do this, one way is undetermined coefficients, have you heard of that before?

idealist10 · Answer

Yes. But in order to do the undetermined coefficients way, I need to form the initial guess yp.

kainui · Answer

Yeah good, I think becoming good at guesses is just something you get through experience. Make a guess and I'll help you see why it's a good or bad guess.

idealist10 · Answer

$$y _{p}=\cos2x+\sin2x$$

kainui · Answer

Yeah, that's good, the only thing you need are coefficients to be determined on each term so keep them separate, $$y_p = A \cos 2x + B \sin 2x $$

idealist10 · Answer

So $$y _{p}=Acos2x+Bsin2x$$

idealist10 · Answer

But I didn't get the right answer. The answer is $$y _{p}=x(\cos2x-3\sin2x).$$

kainui · Answer

This happens from time to time and there's not much you can really do to predict it, However when you do get your answer after your guess, you need to check that you have a linearly independent set of solutions. Have you heard of the Wronskian?

If it turns out that your solutions are linearly dependent then you need to make another guess, usually you can just throw a t on your old solution or guess and work it through so that you do end up with linearly independent solutions.

idealist10 · Answer

Okay, thank you very much for the help.

irishboy123 · Answer

if you'd ripped out the complementary solution first, you would have predicted that 

$y'' + 4y = 0$ leads to $y_c = A \sin 2x + B \cos 2x$

thomas5267 · Answer

You need to guess $y_p=x(C\sin(2x)+D\cos(2x))$ because the complementary solution is $y_c=A\sin(2x)+B\cos(2x)$.