Question

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

Answer 1

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

Answer 2

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

Answer 3

use the undetermined coefficients method

Answer 4

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?

Answer 5

What is the undetermined coefficients method?

Answer 6

Oh, i believe that is what i am doing

Answer 7

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

Answer 8

thats what i tried, but i get C = 0

Answer 9

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

Answer 10

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

Answer 11

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

Answer 12

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

Answer 13

try\[y = Cxe^{4x}\]it works

Answer 14

Just trying that now...

Answer 15

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 =)

Answer 16

Oh and thanks heaps... =)

Answer 17

lol, you are welcome