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

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Kelumptus
 2 years ago
Best ResponseYou've already chosen the best response.0\[\frac{ d ^{2}y }{ dx ^{2} }6\frac{ dy }{ dx }+8y=8e^{4x}\]

Kelumptus
 2 years ago
Best ResponseYou've already chosen the best response.0I know that the complimentary function is \[Ae^{4x}+Be^{2x}\] however i seem to arrive at an incorrect answer for the particular integral.

exraven
 2 years ago
Best ResponseYou've already chosen the best response.1use the undetermined coefficients method

Kelumptus
 2 years ago
Best ResponseYou've already chosen the best response.0For 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 \[e^{4x}(16C24C+8C)=8e^{4x}\] 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?

Kelumptus
 2 years ago
Best ResponseYou've already chosen the best response.0What is the undetermined coefficients method?

Kelumptus
 2 years ago
Best ResponseYou've already chosen the best response.0Oh, i believe that is what i am doing

exraven
 2 years ago
Best ResponseYou've already chosen the best response.1your should assume the particular\[y = Ce^{kx}\]then you solve for C and k

Kelumptus
 2 years ago
Best ResponseYou've already chosen the best response.0thats what i tried, but i get C = 0

Kelumptus
 2 years ago
Best ResponseYou've already chosen the best response.0I believe 'k' is actually 4 in this instance.

Kelumptus
 2 years ago
Best ResponseYou've already chosen the best response.0Can you see where i went wrong in working through the answer? Or do you think it is an error in my book?

exraven
 2 years ago
Best ResponseYou've already chosen the best response.1wait I am working on it too, if Ce^(kx) won't work, try Cxe^(kx)

Kelumptus
 2 years ago
Best ResponseYou've already chosen the best response.0Oh... i'll try that only i thought you only did that when there was already a \[e^{kx}\] on the LHS.

exraven
 2 years ago
Best ResponseYou've already chosen the best response.1try\[y = Cxe^{4x}\]it works

Kelumptus
 2 years ago
Best ResponseYou've already chosen the best response.0Just trying that now...

Kelumptus
 2 years ago
Best ResponseYou've already chosen the best response.0Awesome, 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 =)

Kelumptus
 2 years ago
Best ResponseYou've already chosen the best response.0Oh and thanks heaps... =)
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