Solve given DE by undetermined coefficients Superposition Ap

Solve given DE by undetermined coefficients (Superposition Approach) y\"+y=2x sinx

Solution

The aux equation is m^2+1-0

and hence general solution is

y = A cos x+Bsin x

Particular solution by guess will be of the form

c1x sin x + c2 cosx

Hence y = A cos x+Bsin x+c1x sin x + c2 cosx

y\' = -A sinx + Bcosx+c1x cosx +c1sinx _c2 sinx

  

y\" =- Acos x-Bsin x +c1cosx-c1 xsinx +c1cosx-c2 cosx

   = -y + c1cosx-c1 xsinx +c1cosx-c2 cosx

Compare with y\"+y = 2xsin x

We get c1 = -2 and c2 = c1

Hence solution is

y = A cos x+Bsin x-2x sinx