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Boundary value problem

  1. Dec 8, 2009 #1
    1. The problem statement, all variables and given/known data

    a) Solve for the BVP. Where A is a real number.
    b) For what values of A does there exist a unique solutions? What is the solution?
    c) For what values of A do there exist infintely many solutions?
    d) For what values of A do there exist no solutions?


    2. Relevant equations

    y'' + y = A + sin(2x)
    y(0) = y'([tex]\pi[/tex]/2) = 2

    3. The attempt at a solution


    y = yh + yp
    0 = 1+ [tex]\lambda[/tex]2
    yh =c1*cos(x) + c2*cos(x)

    yp = A + B*sin(2x)
    y = A + B*sin(2x)
    y'' = -4B*sin(2x)
    A + sin(2x) = A -3B*sin(2x)
    A = A, B = -1/3
    yp = A - 1/3*sin(2x)


    y = A - 1/3*sin(2x) + c1*cos(x) + c2*sin(x)
    y' = - 2/3*sin(2x) - c1*sin(x) + c2*cos(x)
    2 = A + c1
    2 = 2/3 + c2
    c2 = 8/3

    y = A - 1/3*sin(2x) + c1*cos(x) + 8/3*sin(x)

    I'm confused about answering the questions. A would be equal to all real numbers, since one could solve for c1. How can I give the solution? There is a unique solution for each value of A, which I would have to write infinte solutions. And there is no value of A when there is an inifiite amount of solutions or no values.

    IF A is defined, what would the answer be?
     
  2. jcsd
  3. Dec 9, 2009 #2

    HallsofIvy

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    So c1= 2-A

    y= A- 1/2 sin(2x)+ (2- A)cos(x)+ 8/3 sin(x)

     
  4. Dec 9, 2009 #3
    So:
    y= A- 1/2 sin(2x)+ (2- A)cos(x)+ 8/3 sin(x)

    c) For what values of A do there exist infintely many solutions?
    d) For what values of A do there exist no solutions?

    Would c and d then be no values of A have infintely many solutions or nonexistent solution?
     
  5. Dec 10, 2009 #4

    HallsofIvy

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