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2nd Order DE

  1. Nov 20, 2013 #1
    Mod note: Reinstated problem after poster deleted it.
    1. The problem statement, all variables and given/known data
    Just wondering if I did this correctly: ##y''+4y'+4y=e^{x}## and initial conditions ##y(0)=0; y'(0)=1##

    2. Relevant equations

    3. The attempt at a solution
    So I found the characteristic equation to be ##r^{2}+4r+4=0## so r=-2 and the general solution is then: ##y_{g}=c_{1}e^{-2x}-c_{2}xe^{-2x}## and particular solution: ##y_{p}=Ae^{x}## and obviously the first and second derivatives are going to be the same thing. So plugging the particular solution into the problem: ##Ae^{x}+4Ae^{x}+4Ae^{x}=e^{x}##, so ##A=\frac{1}{9}##.

    Now ##y=y_{g}+y_{p}## Which is: ##y=c_{1}e^{-2x}+c_{2}xe^{-2x}+\frac{1}{9}e^{x}##. Finally taking initial conditions y(0)=0 y'(0)=1 I get: $$y=\frac{-1}{9}e^{-2x}+\frac{-7}{9}xe^{-2x}+\frac{1}{9}e^{x}$$
    Last edited by a moderator: Nov 21, 2013
  2. jcsd
  3. Nov 20, 2013 #2


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    You can check these things yourself. Substitute your solution into the DE and conditions you are given. That's what the checkers you enlisted will do. I agree that y satisfies the DE and that y(0)=0. I don't think y'(0)=1.
  4. Nov 20, 2013 #3
    I'm not so much concerned with the answer, I want to know if I'm doing the work correctly and whether or not there's a simpler way to solve it.
  5. Nov 20, 2013 #4


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    You are doing it correctly and I don't think there's simpler way. You just goofed up a little in solving for c1 and c2 in applying the boundary conditions.
  6. Nov 20, 2013 #5
    OK that's all I was wondering thanks.
  7. Nov 20, 2013 #6


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    Now why did you delete the OP? Part of the use of Physics Forums is to provide a resource for people to look up past solutions and get some hints for their own problem. Deleting parts of threads makes them unreadable. That's, in part, why I quoted you. That makes the deletion doubly pointless.
  8. Nov 21, 2013 #7


    Staff: Mentor

    iRaid, For the reason that Dick gave, please don't delete your post just because you got an answer.
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