How to solve this integral equation? (if possible using functional analysis)

Question

I have the following integral equation: $$u(x)=1+2 \int_{0}^{\pi} u(t) \cos (x+t) d t,$$ and I need to solve it for $u$, this is an exercise from my course of functional analysis, but I have never seen before an integral equation, and I have no idea how to approach this. Can you help me here?

Answer 1

You have $$u''(x) = 2 \int_{0}^{\pi} u(t) \frac{\partial^2}{\partial x^2} \cos (x+t) dt =-2 \int_{0}^{\pi} u(t) \cos (x+t) dt =-u(x)+1$$

$$\frac{d^2}{dx^2}(u(x)-1) = -u(x)+1 $$ The solution will be $$u(x) = 1 + a \sin(x)+b\cos(x)$$ with $a,b\in \mathbb{R}$