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Help with Initial Value Problems

  1. Nov 4, 2004 #1
    Hi, I have a Calculus test coming up and am very confused on how to start approching these types of problems (initial value problems) If possible can someone work these 2 different types of problems out and show some work so I know how to proceed with the rest of them? I don't think they are that hard, for some reason I'm just confused. Thanks


    Problem 1: y' + 5y = t, y(0) = 0

    Problem 2: y' = y/sqrt(t), y(1) = -1

    Thanks again for all those who help!
     
  2. jcsd
  3. Nov 4, 2004 #2
    Problem 1: We first find a particular solution to y' + 5y = t. Suppose y_p = at + b, then the equation becomes

    a + (5at + 5b) = t

    <=>

    { 5a = 1
    { a + 5b = 0

    <=>

    { a = 1/5
    { b = -1/25

    Thus y_p = t/5 - 1/25 is a particular solution.

    We must then solve the homogenous equation:

    y' + 5y = 0.

    This is a rather well-studied equation, y_h = Ce^(-5t) is the general solution (it could be solved with an integrating factor also). Thus, the general solution to y' + 5y = t can be written:

    y = y_p + y_h = t/5 - 1/25 + Ce^(-5t). You can then use your initial condition to calculate C.

    Problem 2: y' = y/sqrt(t) <=> dy/dt = y/sqrt(t) <=> 1/y * dy = 1/sqrt(t) * dt. What happens if you integrate both sides?
     
  4. Nov 5, 2004 #3

    arildno

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    Just adding to Muzza here:
    Problem 1)
    If you know about the "integrating factor"-technique, you may use this technique instead to derive the solution.
     
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