# I Fredholm integral equation with separable kernel

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1. Jul 9, 2017

### Jianphys17

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. Jul 9, 2017

### Haborix

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.

3. Jul 10, 2017

### Jianphys17

Yes, If you can kindly help me understand how to proceed to solve it !

4. Jul 10, 2017

### Haborix

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.

5. Jul 10, 2017

### Jianphys17

Sorry, the book It gives me the solution, but I do not know how to proceed...

6. Jul 10, 2017

### Haborix

Did you try integrating the equation as I suggested?

7. Jul 13, 2017

### Jianphys17

Sorry, i've been absent for a few days.. anyway yes, but how?

8. Jul 13, 2017

### Haborix

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 $x$.