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Differential Equations 2nd Order Linear Constant-coefficient Problem

  1. Nov 29, 2011 #1
    1. The problem statement, all variables and given/known data
    solve:
    160y''=160g-ky
    y(0)=-200 and y'(0)=0

    2. The attempt at a solution

    I tried to use guess and check to solve this equation, but it didn't turn out nice at all...

    y''=9.8 - (ky)/160
    y''+(ky)/160 = 9.8, guess y=e^(λt), y'=λe^(λt), y''=λ^2e^(λt) : this gives

    e^(λt)*(λ^2+(k/160))=9.8

    Every problem that I've done has not had a term without a y'',y',and a y, so I have no idea where to go from here, or alternative methods to solving this problem.
     
  2. jcsd
  3. Nov 29, 2011 #2

    Mute

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    Homework Helper

    This is a linear, second order non-homogeneous problem. The general solution to a non-homogeneous problem looks like y(t) = yh(t) + Yp(t), where yh(t) is the general solution to the homogeneous problem (i.e., a problem where some function of y and its derivatives = 0, and not some constant or other function) and Yp(t) is the particular solution for the inhomogeneous part.

    When you start by trying a solution of the form y(t) = exp(λt), that's good, but it will only help you with the homogeneous problem 160 y'' + ky = 0. Solving that problem will give you the homogeneous solution. Then, can you guess what kind of particular solution you should add to that to get the full inhomogeneous solution?
     
  4. Nov 29, 2011 #3

    ehild

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    Gold Member

    Have you learnt about differential equations? This is an inhomogeneous linear equation: the term without y makes it inhomogeneous. You get the solution if you solve the homogeneous equation first (that which contains y, y', y" only): 160y"+ky=0, than add a particular solution of the original equation. You get that particular solution if this equation if you assume that y"=0.

    ehild

    EDIT: Mute beat me...
     
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