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anonymous
 5 years ago
Solve the ODE
y''''3y'''+4''=2(x1)e^x
anonymous
 5 years ago
Solve the ODE y''''3y'''+4''=2(x1)e^x

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anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0http://www.wolframalpha.com/input/?i=y%27%27%27%273y%27%27%27%2B4y%27%27+%3D+2%28x1%29e^x

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0lol ya ok... could've done that... need the work

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0This is a massive question to do online. I'll do as much as I can...

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Make the substitution \[v=y'\]Then your equation becomes\[v''3v'+4v=e^x(2x2)\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0You need to solve the homogeneous equation first, then find the particular solution.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0\[v''3v'+4v=0\]has characteristic equation (assuming a solution \[v=e^{{\lambda}x}\])\[{\lambda}^23{\lambda}+4=0\rightarrow {\lambda}=\frac{3}{2}{\pm}\frac{\sqrt{7}}{2}i\]which yields a homogeneous solution,\[v=c_1e^{3x/2}\cos(\frac{\sqrt{7}x}{2})+c_2e^{3x/2}\sin(\frac{\sqrt{7}x}{2})\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0The particular solution can be found by attempting a trial solution based on the RHS of the DE, so attempt\[v=e^x(ax+b)\]When you substitute this into the DE, you end up with the following\[2ax+(2ba)=2x2\](the exponentials cancel). This is true only for a=1 and b=1/2, so the particular solution is\[v=e^x(x\frac{1}{2})\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Your total solution is therefore,

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0\[v=c_1e^{3x/2}\cos(\frac{\sqrt{7}}{2}x)+c_2e^{3x/2}\sin(\frac{\sqrt{7}}{2}x)+e^x(x\frac{1}{2})\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Now, since \[v=\frac{dy}{dx}\] you have to integrate v to find y...this I shall leave to you...if you need further assistance, let me know. Good luck.
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