2nd order non homogeneous equation

[mitch987]

Homework Statement

y'' + y = -2 Sinx

Homework Equations

The Attempt at a Solution

finding the homogeneous solution, is simple;
yh(x) = C1 Cos(x) + C2 Sin(x)

for the particular solution,
I let y = A Cos(x) + B Sin(x)
thus, y' = -A Sin(x) + B Cos(x)
y'' = -A Cos(x) - B Sin(x)

substituting these into the differential equation;
-A Cos(x) - B Sin(x) + A Cos(x) + B Sin(x) = -2 Sin(x)
0 = -2 Sin(x)

This is where i get stuck as I'm unable to find any values for A & B and hence, cannot find the particular solution in order to find the general solution y(x) = yh(x) + yp(x)
Any help would be greatly appreciated.
Thanks,

 

[fzero]

The particular solution must be linearly independent of the homogeneous solutions. Since those are already $$\sin x$$ and $$\cos x$$, you should try $$x\sin x$$ and $$x\cos x$$.

 

[mitch987]

of course! Forgot that the particular solution cannot equal the homogeneous solution.
Thanks for your help.