Which of the following satisfies the equation $\phi(x)=f(x)+\int_{0}^{x}\sin(x-t)\phi(t)dt$

Question

Let $\phi$ satisfy $$\phi(x)=f(x)+\int_{0}^{x}\sin(x-t)\phi(t)dt.$$ Then $\phi$ is given by

1. $\phi(x)=f(x)+\int_{0}^{x}(x-t)f(t)dt.$
2. $\phi(x)=f(x)-\int_{0}^{x}(x-t)f(t)dt.$
3. $\phi(x)=f(x)-\int_{0}^{x}\cos(x-t)f(t)dt.$
4. $\phi(x)=f(x)-\int_{0}^{x}\sin(x-t)f(t)dt.$

So I try to solve the problem by calculating the separated kernel $\sin(x-t)=\sin x \cos t-\cos x \sin t$ and trying to find the eigenvalues $\lambda$ but it ended up in a messy equation which I couldn't solve further. So how can I do this ? Any hints or help will be great. Thanks.

Answer 1

Hint: Try to convert it into an ordinary differential equation which must be $\phi''(x)=f''(x)+f(x)$. Now, can you integrate it twice to get option $1$.