## Kelumptus 2 years ago I am having a problem in arriving at the correct answer to a second order inhomogeneous differential equation (below):

1. Kelumptus

$\frac{ d ^{2}y }{ dx ^{2} }-6\frac{ dy }{ dx }+8y=8e^{4x}$

2. Kelumptus

I know that the complimentary function is $Ae^{4x}+Be^{2x}$ however i seem to arrive at an incorrect answer for the particular integral.

3. exraven

use the undetermined coefficients method

4. Kelumptus

For the particular integral i get $16Ce^{4x}-24Ce^{4x}+8Ce^{4x}=8e^{4x}$ Based on derivatives of the general function $Ce^{4x}$ I therefore get $e^{4x}(16C-24C+8C)=8e^{4x}$ Therefore $0=8$ Which means there is no solution. The answer in my book for the particular integral however is $4e^{4x}$. Where did i go wrong?

5. Kelumptus

What is the undetermined coefficients method?

6. Kelumptus

Oh, i believe that is what i am doing

7. exraven

your should assume the particular$y = Ce^{kx}$then you solve for C and k

8. Kelumptus

thats what i tried, but i get C = 0

9. Kelumptus

I believe 'k' is actually 4 in this instance.

10. Kelumptus

Can you see where i went wrong in working through the answer? Or do you think it is an error in my book?

11. exraven

wait I am working on it too, if Ce^(kx) won't work, try Cxe^(kx)

12. Kelumptus

Oh... i'll try that only i thought you only did that when there was already a $e^{kx}$ on the LHS.

13. exraven

try$y = Cxe^{4x}$it works

14. Kelumptus

Just trying that now...

15. Kelumptus

Awesome, as you said, that works. I didn't realize that you could use that method when there is no solution of a coefficient. I thought it was only when an element of the general solution appeared in the LHS. Good to know, i've learned a new trick =)

16. Kelumptus

Oh and thanks heaps... =)

17. exraven

lol, you are welcome