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I Fredholm integral equation with separable kernel

  1. Jul 9, 2017 #1
    Hi at all
    On my math methods book, i came across the following Fredholm integ eq with separable ker:

    1) φ(x)-4∫sin^2xφ(t)dt = 2x-pi
    With integral ends(0,pi/2)
    I do not know how to proceed, for the solution...
     
  2. jcsd
  3. Jul 9, 2017 #2
    Is this an assigned problem?

    Could you verify that the equation you are working with can be written as follows: $$\phi(x)-4\sin^2(x)\left(\int_0^{\frac{\pi}{2}}\phi(t) dt\right)=2x-\pi.$$ I want to be sure I'm not missing any hidden dependence on the variables involved.
     
  4. Jul 10, 2017 #3
    Yes, If you can kindly help me understand how to proceed to solve it ! :bow:
     
  5. Jul 10, 2017 #4
    It's clear that if we knew the value of ##\int\phi dt##, then this would just be an algebra problem. Think about what integrating both sides of the equation from ##0## to ##\pi/2## would allow you to do.
     
  6. Jul 10, 2017 #5
    Sorry, the book It gives me the solution, but I do not know how to proceed...:olduhh:
     
  7. Jul 10, 2017 #6
    Did you try integrating the equation as I suggested?
     
  8. Jul 13, 2017 #7
    Sorry, i've been absent for a few days.. anyway yes, but how?
     
  9. Jul 13, 2017 #8
    I'm going to write out explicitly what I think you should compute, but I think you should be the one to perform the computation.

    $$
    \int_0^{\frac{\pi}{2}}\left(\phi(x)-4\sin^2(x)\left(\int_0^{\frac{\pi}{2}}\phi(t) dt\right)\right)dx=\int_0^{\frac{\pi}{2}}\left(2x-\pi\right)dx
    $$

    Remember that ##\int \phi(t) dt## is just a number; it does not depend on ##x##.
     
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