elica85 diff eq..determine the form of a particular solution for the given nonhomogeneous equation y''+2y'=3-4t 2 years ago 2 years ago

1. amistre64

yp = some educated guess yp=At + B ; if these are not already a linear solution to the homogenous part ...

2. elica85

for fundamental solution set, i have y1=1, y2=e^(-2t) f(t)=3-4t polynomial method so yp=-At+B...check to see if any matches the fund.set. it does so... yp=-At^2+Bt find y'p and y''p and plug in. now solving for A and B is where i'm stuck

3. amistre64

then lets work thru it so I can catch up :)

4. elica85

ok thx

5. amistre64

r^2+2r=0 r=0,-2 right?

6. elica85

at this point, it's just algebra which apparently, i'm very weak at

7. elica85

right

8. amistre64

$y=c_1e^{0x}+c_2e^{-2x}+y_p$ $y=c_1+c_2e^{-2x}+y_p$

9. amistre64

yp = At + B ; but we already have a constant so we need to up the ante yp = At^2 + Bt ; and derive

10. amistre64

yp = At^2 + Bt yp'= 2At + B yp''= 2A agreed?

11. amistre64

yp''+2yp'=3-4t (2A) + 2(2At + B)=3-4t im going to do this in lowercase, just easier to type for me 2a +4at +2b = 3 -4t (4a)t = -4t ; a=-1 (2a+2b) = 3 -2+2b = 3 ; b=5/2

12. elica85

ah ok, i almost did that to solve for a but wasn't sure if it was correct, thank you

13. amistre64

at^2 + bt = -t^2+ 5/2 t $y=c_1+c_2e^{-2x}-t^2+\frac 52 t$