Method of Undetermined Coefficients

[Chemmjr18]

Homework Statement

Find a particular solution yp of the given equation:
y''+y=sinx+xcosx

Homework Equations

The Attempt at a Solution

1) First I got the temporary y_p: Asinx+Bcosx+(Cx+D)sinx+(Ex+F)cosx

2) Then I got the associated homogenous eq. check for to check for duplicates: C₁cosx+C₂sinx

3) Since there were duplicates, I multiplied the temporary solution by x: Axsinx+Bxcosx+(Cx²+Dx)sinx+(Ex²+Fx)cosx

4) I obtained the 2nd derivative of this and refined it: (2A+2C+2F)cosx+(-2B-2D+2E)sinx+(4E-D-B)xcosx+(-A-4C-F)xsinx-Cx²cosx-Ex²sinx

5) I then added this to 2) to get (2A+2C+2F)cosx+(-2B-2D+2E)sinx+(4E)xcosx+(-4C)xsinx=sinx+xcosx

I have more unknowns than I do equations, so I can't solve. Where'd I go wrong? I've been combing through the problem for almost 4 hours...

 

[Chemmjr18]

Homework Statement

Find a particular solution yp of the given equation:
y''+y=sinx+xcosx

Homework Equations

The Attempt at a Solution

1) First I got the temporary y_p: Asinx+Bcosx+(Cx+D)sinx+(Ex+F)cosx

2) Then I got the associated homogenous eq. check for to check for duplicates: C₁cosx+C₂sinx

3) Since there were duplicates, I multiplied the temporary solution by x: Axsinx+Bxcosx+(Cx²+Dx)sinx+(Ex²+Fx)cosx

4) I obtained the 2nd derivative of this and refined it: (2A+2C+2F)cosx+(-2B-2D+2E)sinx+(4E-D-B)xcosx+(-A-4C-F)xsinx-Cx²cosx-Ex²sinx

5) I then added this to 2) to get (2A+2C+2F)cosx+(-2B-2D+2E)sinx+(4E)xcosx+(-4C)xsinx=sinx+xcosx

I have more unknowns than I do equations, so I can't solve. Where'd I go wrong? I've been combing through the problem for almost 4 hours...

Nevermind, I figured it out.