Find $f(x)=?$ functional equation

Question

I would appreciate if somebody could help me with the following problem:

Q: Find $f(x)$ ($f'(x)$: conti-function , $x \in\mathbb{R}$) $$f(x)=\sin ^2x+\int_{0}^{x}tf(t)dt$$

Answer 1

Differentiate both sides wrt x to get $$f'(x)=2\sin(x)\cos(x)+xf(x)$$ which is a linear ODE in f(x). Then use the integrating factor method. You'll get f(x) down to an integral in terms of x, but it's not going to be an elementary antiderivative.

Answer 2

$f '(x)-xf(x)=2\sin(x)\cos(x)$ $\to$ $$f(x)=e^{\left(-\dfrac{x^2}{2}\right)} \left(\int_0^x e^{\left(-\dfrac{t^2}{2}\right)}\cdot2\sin(t)\cos(t)dt+c\right)$$

Answer 3

It's basically a fundamental theorem of calculus problem. I was thinking your only confusion would probably be how to get rid of $t$. Remember the fundamental theorem of calculus says if $F(x)=\int_0^t f(t)dt$, then we have $F'(x)=f(x)$. I think that's probably the only trick you need to use. The rest of this question is just standard ODE solving.

Please don't select my answer. It's just meant to complement other answers.