# Second order non homogeneous diff. equation at constant coefficients

• May 3rd 2011, 08:49 AM
Ulysses
Second order non homogeneous diff. equation at constant coefficients
Hi there. I had some trouble trying to solve this:

$y''+y=\cos x +3\sin \color{red}2x$ (1)

At first I just found the solution for the homogeneous equation:
$y_h=e^{\lambda x} \rightarrow \lambda^2+1=0 \rightarrow \lambda_1,\lambda_2=\pm i$

Then $y_h=C_1\cos x+ C_2 \sin x$

So I've tried to find the particular solution. I thought I should suggest an equation like:
$y_p=A\cos x +b\sin x+ C\cos 2x +D\sin 2x$
$y_p'=-A\sin x +B\cos x - 2C\sin 2x + 2D\cos 2x$
$y_p''=-A\cos x -B\sin x - 4C\cos 2x - 4D\sin 2x$
But then, when I've tried to find the undetermined coefficients for $y_p$ coefficients replacing in (1):

$-A\cos x -B\sin x - 4C\cos 2x - 4D\sin 2x+A\cos x +b\sin x+ C\cos 2x +D\sin 2x=\cos x +3\sin x \rightarrow -3C\cos 2x -3D\sin 2x=cos(x)+3\sin 2x$
Then A=0,B=0,C=0 and D=-1.

So the general solution should be
$y(x)=y_h+y_p=C_1\cos x+ C_2 \sin x-\sin 2x$
But with wolfram alpha I've corroborated my solution is wrong: y&#39;&#39;&#43;y&#61;cos&#40;x&#41;&#43;3 sin&#40;2x&#41; - Wolfram|Alpha

So, where is the mistake and how should I do this?

I think I've found unless one mistake, just noted it. I should use $y_p=Ax \cos x +B x \sin x + C\sin 2x+ D\cos 2x$ instead of $y_p=A\cos x +b\sin x+ C\cos 2x +D\sin 2x$, because the first two terms are linearly dependent with the homogeneous solution, right?
• May 3rd 2011, 08:55 AM
Ackbeet
Don't use any terms like $\cos(2x).$ You're just creating unnecessary work for yourself. Do multiply the homogeneous solution by powers of x, though, to get linearly independent solution candidates. I think you'll only need one power of x. If that doesn't work out, try two.
• May 3rd 2011, 09:12 AM
Ulysses
I'm sorry, I've made a mistake* in (1). It was 2x instead of x. I think I need the additional terms. Am I right?

*Just corrected.
• May 3rd 2011, 11:39 AM
Ackbeet
Try

$y_{p}=Ax\cos(x)+Bx\sin(x).$