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## Homework Statement

y''+4y=t^2+3e^t

y(0)=0

y'(0)=2

## Homework Equations

CE: r^2+4

r=+/-2i

gs: y=c1 cos(2t) + c2 sin(2t)

## The Attempt at a Solution

guess:

yp=(At^2+Bt+C)e^t

yp'=At^2e^t+2Ate^t+Bte^t+Be^t+Ce^t

yp''=At^2e^t+4Ate^t+Bte^t+2Ae^t+2Be^t+Ce^t

back into problem:

At^2e^t+4Ate^t+Bte^t+2Ae^t+2Be^t+Ce^t+4At^2e^t+4Ce^t=t^2+3e^t

which becomes:

5At^2e^t=t^2 then becomes: A=e^-t/5

then

4At^et+Bte^t+4Bte^t=0 which becomes: 4A+B+4B=0, B=-(4e^-t)/25

then:

2Ae^t+2Be^t+Ce^t+4Ce^t=3e^t, e^t's cancel which becomes:

2A+2B+5C=3 ---> C=[(75e^t)-2)e^-t]/125

ok, i'm sure i'm not doing this correctly. the book answer is:

7/10sin(2t)-19/40cos2t+1/4t^2-1/8+3/5e^t

what am i doing wrong? am i skipping a step?