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Differential Equation Resonance

  1. Jul 8, 2017 #1
    1. The problem statement, all variables and given/known data
    I was reading a PDE book with a problem of resonance
    $$
    y_{tt} (x,t) = y_{xx} (x,t) + A \sin( \omega t)
    $$
    After some work it arrived to a problem of variation of parameters for each odd eigenvalue. To solve it, it uses
    $$
    y''(t)+a^{2} y(t) = b \sin ( \omega t) \qquad y(0)=0 \quad y'(0)=0
    $$
    has the solution
    $$
    y(t) = \dfrac{ b }{ \omega^{2} - a^{2} } \left( \frac{ \omega }{a} \sin (at) - \sin(wt) \right)
    $$
    I would like to solve this, but as using the method of undetermined coefficients feels like guessing (and for that I can simply verify the solution, which I have already done) I tried to solve it using variation of parameters.

    The thing is, I got a real mess, and after a lot of simplifications with trigonometry I arrived to an expression that's closer to that but apparently I still need to do more work.
    I wanted to know if there's a more simple way to solve it without using the method of undetermined coefficients or it's really messy if you avoid that.
    The equation I arrived to is
    $$
    y(t) = \dfrac{ b }{ \omega^{2} - a^{2} }
    \left( \omega \cos( x \omega) \sin( x \omega) \cos(xa)
    +a \sin(x \omega)^{2} \sin (xa)
    +a \cos(x \omega) \sin(x \omega) \cos (xa) - \omega \sin (xa) \cos ( x \omega )^{2} \right) + c \sin (x \omega )
    $$
    Where c has to be determined after putting initial conditions, I think that's ##(-1)( \omega a)##. Anyway, I probably made a mistake in a previous step, and simplying this seems too much work, so I guess there's an easier way to do this.
    PD: I have no idea why this doesn't "compile".
     
  2. jcsd
  3. Jul 8, 2017 #2
    Working a little more I've got
    $$
    y(t) = \frac{1}{2} \dfrac{ b }{ \omega^{2} - a^{2} }
    \bigg( a \sin ( ( \omega+a) x ) \sin (x \omega) + \omega \sin ( (\omega-a) x ) \cos (x \omega) \bigg) + c \sin (x \omega )
    $$
    I probably made a mistake in a previous step.
     
  4. Jul 8, 2017 #3

    Ray Vickson

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    A quicker and easier way in this case is to use Laplace transforms. The Laplace transform ##Y(s)## of ##y(t)## satisfies the equation
    $$ (s^2+a^2) Y(s) = \frac{\omega b}{\omega^2+s^2} $$
    You can solve for ##Y(s)##, express the result using partial fractions, and then just invert two transforms of the form ##A/(s^2+a^2)## and ##B/(s^2+\omega^2)## for some constants ##A,B##. Both of these are standard, off-the-shelf inversions. This gives you the desired form immediately, with no need for any further simplification.
     
  5. Jul 8, 2017 #4

    vela

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    You could also find the impulse response ##h(t)##, which satisfies
    $$h''(t)+a^2h(t) = \delta(t)$$ and the appropriate conditions on ##h## and ##h'## at ##t=0##. To obtain the solution, convolve ##h## with ##b \sin \omega t##:
    $$y(t) = \int_0^t b\sin\omega\tau\,h(\tau-t)\,d\tau.$$
     
  6. Jul 8, 2017 #5
    Oh thank you! I didn't use the Laplace transform since I finish my ODE course. This is going to be fun :D
     
  7. Jul 8, 2017 #6

    Ray Vickson

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    You could instead use the method outlined by Vela in #4, which is really tantamount to finding a "Green's function" for the problem.

    My personal ranking of the methods, in descending order of liking, is
    Laplace --> Green's function --> Undetermined coefficients --> Variation of parameters.
    Of course, Laplace is only practical for constant-coefficient linear DEs; the other methods are more generally applicable.
     
  8. Jul 9, 2017 #7
    Oh. I'm self studying PDE, I have been given a book to follow but it has too few topics.
    A good book to learn about green's functions? Is Evan's one recommended?
    Laplace transform was very good, even for the case ##a= \omega ##
     
  9. Jul 9, 2017 #8

    Ray Vickson

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    Sorry: cannot help much here. I took that material about 55 years ago in an undergraduate course on "mathematical methods in physics". I don't remember the textbook (which might be out of print anyway) and I have not kept up with the books available.
     
  10. Jul 9, 2017 #9

    WWGD

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    Maybe you can go over the Math- and Engineering Physics books in your school's ( or local*) library, browse through them and see which ones you like.


    * Assuming your local library has Math-related books.
     
  11. Jul 9, 2017 #10
    I learned about Greens functions from Arfkin's book "Mathematical Methods for Physicists" in grad school, but that was 30 years ago. Books are expensive. There are plenty of free online resources dealing with Greens functions. Google is your friend.
     
  12. Jul 9, 2017 #11
    My university library has almost no math books (I think it has 3 different books on differential equations and 1 on metric topology and that's the most advanced thing it has), but where I live there's an institute where they have engineering and physics undergrad and graduate students, it has a large and nice library that I can have access to. So I might go to that place to read for a while.
    In my university I was the only math student for, like 6 courses and I had to selfstudy some of those bymyself :p, so you can imagine how much resources they have hehe. Considering the difference in resourses I might have made a mistake chosing to study math instead of physics.
     
  13. Jul 9, 2017 #12

    WWGD

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    Wow, you must be way in the boondocks.
     
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