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February 7th, 2018, 09:56 AM  #1 
Newbie Joined: Feb 2018 From: Philippines Posts: 2 Thanks: 0  Differential Equation Problem
Can somebody help me with this differential equation? Thank you very much y''' + y'' + y = x^2 + 3e^3x 
February 7th, 2018, 10:32 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,207 Thanks: 1160 
mathematica returns an absurdly long answer. are you sure there are no typos? 
February 7th, 2018, 08:33 PM  #3 
Banned Camp Joined: Apr 2017 From: durban Posts: 22 Thanks: 0 Math Focus: Algebra 
hmmm

February 7th, 2018, 09:24 PM  #4 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,515 Thanks: 2515 Math Focus: Mainly analysis and algebra 
The principal problem is that the characteristic (homogeneous) equation doesn't have pleasant roots.

February 8th, 2018, 06:30 AM  #5 
Newbie Joined: Feb 2018 From: Philippines Posts: 2 Thanks: 0  
February 8th, 2018, 10:47 AM  #6 
Global Moderator Joined: Dec 2006 Posts: 19,994 Thanks: 1856 
If the equation is $y'' + y' + y = x^2 + 3e^{3x}$, a particular solution is $y = x^2  2x + \frac{3}{13}e^{3x}$. Add to that the general solution of $y'' + y' + y = 0$. 
February 8th, 2018, 04:26 PM  #7 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894 
The characteristic equation for y''+ y'+ y= 0 is $r^2+ r+ 1= 0$. Writing that as $r^2+ r= 1$ and "completing the square", $r^2+ r+ \frac{1}{4}= (r+ \frac{1}{2})^2= 1+ \frac{1}{4}= \frac{3}{4}$. Taking the square root of both sides $r+ \frac{1}{2}= \pm\frac{\sqrt{3}}{2}i$ and $r= \frac{1}{2}\pm\frac{\sqrt{3}}{2}i$. The general solution to that homogeneous differential equation is $y(x)= e^{x/2}\left(C_1\cos(x\sqrt{3}/2)+ C_2\sin(x\sqrt{3}/2)\right)$ Last edited by skipjack; February 8th, 2018 at 05:56 PM. 

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