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

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

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

exravenBest ResponseYou've already chosen the best response.1
use the undetermined coefficients method
 one year ago

KelumptusBest ResponseYou've already chosen the best response.0
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 \[e^{4x}(16C24C+8C)=8e^{4x}\] 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?
 one year ago

KelumptusBest ResponseYou've already chosen the best response.0
What is the undetermined coefficients method?
 one year ago

KelumptusBest ResponseYou've already chosen the best response.0
Oh, i believe that is what i am doing
 one year ago

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

KelumptusBest ResponseYou've already chosen the best response.0
thats what i tried, but i get C = 0
 one year ago

KelumptusBest ResponseYou've already chosen the best response.0
I believe 'k' is actually 4 in this instance.
 one year ago

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

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

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

exravenBest ResponseYou've already chosen the best response.1
try\[y = Cxe^{4x}\]it works
 one year ago

KelumptusBest ResponseYou've already chosen the best response.0
Just trying that now...
 one year ago

KelumptusBest ResponseYou've already chosen the best response.0
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 =)
 one year ago

KelumptusBest ResponseYou've already chosen the best response.0
Oh and thanks heaps... =)
 one year ago
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