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Toronto Math Forum
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MAT244-2013S
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MidTerm
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MT Problem 4
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Topic: MT Problem 4 (Read 12302 times)
Victor Ivrii
Administrator
Elder Member
Posts: 2607
Karma: 0
MT Problem 4
«
on:
March 06, 2013, 09:09:26 PM »
Find a particular solution of equation
\begin{equation*}
y'''-2y''+4y'-8y=e^{3x}
\end{equation*}
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Branden Zipplinger
Jr. Member
Posts: 7
Karma: 2
Re: MT Problem 4
«
Reply #1 on:
March 06, 2013, 10:04:31 PM »
Here is the solution. I apologize for the picture instead of code, but time was of the essence
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Matthew Cristoferi-Paolucci
Jr. Member
Posts: 10
Karma: 8
Re: MT Problem 4
«
Reply #2 on:
March 06, 2013, 10:07:10 PM »
Heres my solution
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Rudolf-Harri Oberg
Jr. Member
Posts: 9
Karma: 9
Re: MT Problem 4
«
Reply #3 on:
March 06, 2013, 10:12:21 PM »
We use Milman's method:
$$L\left[A\frac{x^m}{m!}e^{rx}\right]=Ae^{rx}\left(Q(r)\frac{x^m}{m!}+Q'(r)\frac{x^{m-1}}{(m-1)!}+Q''(r)\frac{x^{m-2}}{2!(m-2)!}+... \right)$$
In this case $Q=r^3-2r^2+4r-8$. As we see an exponent in the power of three, let $r=3$. We need to evaluate $Q(3)=13$. As there are no polynomial terms, let $m=0$.
Then $L(Ae^{3x})=13Ae^{3x}$ which implies $A=\frac{1}{13}$.
Solution is $Y_p=\frac{1}{13}e^{3x}$.
«
Last Edit: March 06, 2013, 10:16:22 PM by Rudolf-Harri Oberg
»
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Matthew Cristoferi-Paolucci
Jr. Member
Posts: 10
Karma: 8
Re: MT Problem 2a
«
Reply #4 on:
March 06, 2013, 10:18:59 PM »
Im slow today...
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Devangi Vaghela
Jr. Member
Posts: 6
Karma: 3
Re: MT Problem 4
«
Reply #5 on:
March 06, 2013, 10:20:14 PM »
This is my solution!
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Jeong Yeon Yook
Full Member
Posts: 20
Karma: 8
Re: MT Problem 4
«
Reply #6 on:
March 06, 2013, 10:21:30 PM »
solutioin
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Branden Zipplinger
Jr. Member
Posts: 7
Karma: 2
Re: MT Problem 4
«
Reply #7 on:
March 06, 2013, 10:53:05 PM »
there needs to be a correction in my solution: I said r = 2i when its other part of the conjugate pair, -2i, was not mentioned. and I should have subbed the r's with a 1,2, and 3 respectively for good notation
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Victor Lam
Jr. Member
Posts: 13
Karma: 9
Re: MT Problem 4
«
Reply #8 on:
March 07, 2013, 12:11:01 AM »
I'm a lil late
, but here's a quick way to find Y(particular). A slower "coefficients" method can also verify that A = 1/13
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Victor Ivrii
Administrator
Elder Member
Posts: 2607
Karma: 0
Re: MT Problem 4
«
Reply #9 on:
March 07, 2013, 05:00:44 AM »
Guys, are you showing that you have no idea how to scan or to post (Matthew posted it first in the wrong forum)? I decided to distribute several karma points just to encourage those with very few if any
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Toronto Math Forum
»
MAT244-2013S
»
MAT244 Math--Tests
»
MidTerm
(Moderator:
Brian Bi
) »
MT Problem 4