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Homework Help: Perturbation Methods for 1st order ODE - Find the asymptotic solution

  1. Jul 16, 2007 #1
    1. The problem statement, all variables and given/known data

    Apply the regular perturbation method to solve the following ordinary differential equation


    subject to


    Show that the asymptotic solution is of the form


    2. Relevant equations

    http://www.sm.luth.se/~johanb/applmath/chap2en/index.html" [Broken]

    3. The attempt at a solution

    First, I get the base case by setting epsilon to zero.


    Am I supposed to use the "subject to" conditions now?


    I have no idea if that is what I am supposed to do, I hope so otherwise I am completely lost here.

    Now, I assume that a solution is of the form


    Differentiating that approximation


    Expanding the original equation given


    and substituting the approximations into it


    Please tell me if the above substitution and expansion are correct.

    Now I start matching terms according to their order.

    O(1): [tex]\frac{dy_0}{dx}\,=\,-y_0[/tex]

    O([itex]\epsilon[/itex]): [tex]\frac{dy_1}{dx}\,=\,-y_0\,\frac{dy_0}{dx}\,-\,y_1[/tex]

    O([itex]\epsilon^2[/itex]): [tex]\frac{dy_2}{dx}\,=\,-2\,y_0\,\frac{dy_1}{dx}\,-\,y_2[/tex]

    Solving for order one



    OR using the base case that I am unsure about


    But which one is it?

    Thanks in advance for your help!
    Last edited by a moderator: May 3, 2017
  2. jcsd
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