Is there any other solution for $f(x)=\int_0^\pi t f(t)\,\mathrm{d}t+\cos x $? Is there any other solution for $f(x)=\int_0^\pi t f(t)\,\mathrm{d}t+\cos x $? I found one solution as follows but I have no clue how to prove that this solution is unique.
Let $\int_0^\pi t f(t)=\lambda$, we have
\begin{align}
\lambda 
&=\int_0^\pi t f(t) \,\mathrm{d}t\\
&=\int_0^\pi t (\lambda+\cos t)\,\mathrm{d}t\\
&= \frac{4}{\pi^2-2}
\end{align}
Thus $f(x)=\frac{4}{\pi^2-2}+\cos x$.
Question
Is there any other solution? How to prove the uniqueness of this solution?

Assume that you have $f,g$ such that $$\cos(x)=f(x)-\int_0^\pi tf(t)dt=g(x)-\int_0^\pi tg(t)dt.$$ It then follows that $$0=f(x)-g(x)-\int_0^\pi tf(t)dt+\int_0^\pi tg(t)dt=f(x)-g(x)-\int_0^\pi t(f(t)-g(t))dt.$$ So you get $$f(x)-g(x)=\int_0^\pi t(f(t)-g(t))dt=C$$ a constant. So write $f(x)=g(x)+C$ and calculate $$C=\int_0^\pi t(f(t)-g(t))dt=\int_0^\pi t(g(t)+C-g(t))dt\\ =\int_0^\pi t(g(t)-g(t))dt+\int_0^\pi tCdt=C\int_0^\pi tdt =C\frac{\pi^2}{2}.$$
The only value of $C$ such that this equation can hold is $C=0$. Consequently, solutions are unique.
