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February 7th, 2018, 09:56 AM   #1
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Post Differential Equation Problem

Can somebody help me with this differential equation? Thank you very much

y''' + y'' + y = x^2 + 3e^3x
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February 7th, 2018, 10:32 AM   #2
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mathematica returns an absurdly long answer.

are you sure there are no typos?
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February 7th, 2018, 08:33 PM   #3
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hmmm
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February 7th, 2018, 09:24 PM   #4
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The principal problem is that the characteristic (homogeneous) equation doesn't have pleasant roots.
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February 8th, 2018, 06:30 AM   #5
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Quote:
Originally Posted by romsek View Post
mathematica returns an absurdly long answer.

are you sure there are no typos?
Oh, thanks mate! I checked it again. It's supposed to be y" + y' + y
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February 8th, 2018, 10:47 AM   #6
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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 $y'' + y' + y = 0$.
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February 8th, 2018, 04:26 PM   #7
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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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