Fredholm integral equation with separable kernel

[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...

 

[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.

 

[Jianphys17]

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.

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

 

[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.

 

[Jianphys17]

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

 

[Haborix]

Did you try integrating the equation as I suggested?

 

[Jianphys17]

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

 

[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 $x$.