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Basic Initial Value Problems

  1. Mar 10, 2009 #1
    1. The problem statement, all variables and given/known data

    Solve the following IVPs.

    a) y'' + y = 0 for y(x0)=1 and y'(x0)=0

    b) y'' - λ2y = 0 for y(x0)=0 and y'(x0)=1

    2. Relevant equations
    3. The attempt at a solution

    a) Two distinct solutions for y'' + y = 0 are cos(x) and sin(x).

    So, y(x0) = C1cos(x0) + C2sin(x0) = 1

    y'(x0) = -C1sin(x0) + C2cos(x0) = 0

    b) Two distinct solutions for y'' + λ2y = 0 is just e^([tex]\lambda[/tex]x) and e^(-[tex]\lambda[/tex]x)

    So,

    y(x0) = C1e^([tex]\lambda[/tex]x0) + C2e^([tex]\lambda[/tex]x0)) = 0

    y'(x0) = C1λe^([tex]\lambda[/tex]x0) - C2λe^(-[tex]\lambda[/tex]x0) = 1

    Now do I need to go through and solve for the C's or I am OK?
     
  2. jcsd
  3. Mar 10, 2009 #2

    tiny-tim

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    Homework Helper

    Yup! :biggrin:
     
  4. Mar 10, 2009 #3
    Thank you, my notes weren't very clear on the next step.
     
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