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2nd Order Linear Diff. Eqn (homogeneous)

  1. May 5, 2008 #1
    1. The problem statement, all variables and given/known data
    Show that id y = x(t) is a solution of the diff. eqn. y'' + p(t)y' + q(t)y = g(t), where g(t) is not always zero, then y = c*x(t), where c is any constant other than 1, is not a solution.


    2. Relevant equations
    Can someone help me get started?
    Also, since g(t) is not zero, this means that the equation is nonhomogeneous?


    3. The attempt at a solution
     
  2. jcsd
  3. May 6, 2008 #2

    Defennder

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    Yes, this implies the DE is non-homogenous. To show that y = c*x(t) is not a solution just substitute that into the DE. Do you still get g(t) on the RHS?
     
  4. May 6, 2008 #3
    For y = c*x(t): x''(t) + p(t)*x'(t) + q(t)*x(t) = g(t)

    What next?
     
  5. May 6, 2008 #4

    Defennder

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    Note that it is given that y=x(t) is a solution, that means x''(t) + p(t)x'(t) + q(t)x(t) = g(t). The expression you get when you substitute y = cx(t) into the DE is clearly different from this. What does that tell you?
     
  6. May 6, 2008 #5

    HallsofIvy

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    What happened to the c?? If x''(t) + p(t)*x'(t) + q(t)*x(t) = g(t) what is
    (c*x)'' + p(t)*(c*x)' + q(t)*(c*x)?
     
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