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May 26th, 2018, 05:16 AM  #1 
Newbie Joined: May 2018 From: Sri Lanka Posts: 3 Thanks: 0  Differential EquationsHELP
Hey guys, I'm new to here. Attached with this are two problems in a tutorial of our university maths department. Hope you could help me to solve them. Last edited by skipjack; May 26th, 2018 at 07:18 AM. 
May 26th, 2018, 07:32 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 19,722 Thanks: 1806 
(1) (D² + 2D + 1)y = $x \cos x$ Solution: y = (A + B$x$)$e^{x}$ + ($x\sin x  \sin x + \cos x$)/2, where A and B are constants. There are various ways of obtaining that. For example, you can start by using $e^x\!$ as an integrating factor. What method(s) would you be expected to know? Can you confirm that the second question gives the equations below? (2D²  D + 9)y  (D² + D + 3)z = 0 (2D² + D + 7)y  (D²  D + 5)z = 0 They're a bit hard to read. Last edited by skipjack; May 26th, 2018 at 10:36 PM. 
May 26th, 2018, 06:08 PM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,445 Thanks: 2499 Math Focus: Mainly analysis and algebra 
For the second, you can take the sum of the two equations and the difference of the two equations to get the equations \begin{align*}2(D1)(y+z) &= 0 \\ 2(D^2 + 4 )(2yz) &= 0 \end{align*} You can solve these for the two functions $f(x)=y(x)+z(x)$ and $g(x)=2y(x)z(x)$ respectively, and then find $y(x)$ and $z(x)$ as functions of $f(x)$ and $g(x)$. I get \begin{align*} y(x) &= c_1 e^x + c_2 \sin{(2x)} + c_3 \cos{(2x)} \\ z(x) &= 2c_1 e^x  c_2 \sin{(2x)}  c_3 \cos{(2x)} \end{align*} Last edited by v8archie; May 26th, 2018 at 06:40 PM. 
May 26th, 2018, 08:26 PM  #4 
Newbie Joined: May 2018 From: Sri Lanka Posts: 3 Thanks: 0 
thanks alot guys..... 
May 27th, 2018, 05:41 AM  #5 
Newbie Joined: May 2018 From: Sri Lanka Posts: 3 Thanks: 0 
yep.That's the second part.


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