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Problem with finding the complementary solution of ODE

  1. Jun 3, 2012 #1

    Uku

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    Hello!

    On Pauls notes webpage, there is the following problem to be solved by variation of parameters:

    [itex]ty''-(t+1)y'+y=t^2[/itex] (1)
    On the page, the fundamental set of solutions if formed on the basis of the complementary solution. The set is:
    [itex]y_{1}(t)=e^t[/itex] and [itex]y_{2}(t)=t+1[/itex]

    Now, I must be missing something here. Since I get the complementary solution for the homogeneous equation of (1):

    [itex]r=\frac{(t+1)+/- \sqrt{(t+1)^2-4t}}{2t}[/itex] which solves as [itex]r_{1}=1[/itex] and [itex]r_{2}=\frac{1}{t}[/itex] which would give a complementary solution of:

    [itex]Y_{c}=C_{1}e^{t}+C_{2}e^{\frac{1}{t}t}=C_{1}e^{t}+C_{2}e^1[/itex]
    from which I would get [itex]y_{1}(t)=e^t[/itex] and [itex]y_{2}(t)=e[/itex]

    What have I missed, must be simple...

    Regards,
    U.
     
  2. jcsd
  3. Jun 3, 2012 #2

    tiny-tim

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    Hello Uku! :smile:
    no, the characteristic polynomial method only works for constant coefficients,

    not for coefficients which depend on t
     
  4. Jun 3, 2012 #3

    Uku

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    Okay, that is true, thank you. I now read from his example that the set is given by default.

    Still: how would you arrive at [itex]y_{1}[/itex] and [itex]y_{2}[/itex]?

    U.
     
  5. Jun 3, 2012 #4

    tiny-tim

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    dunno :redface:
     
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