Which of the following functions is the solution of the Fredholm integral equation:
$$u(x)+\frac{1}{2}\int_0^1e^{x-t}u(t)dt=2xe^x$$
(A) $u(x)=e^x\big(2x-\frac{2}{3}\big)$
(B) $u(x)=e^x(x^2-x)$
(C) $u(x)=e^x(x-3)$
(D) $u(x)=e^x(3x^2-3x-1)$
My attempt:
I tried substituting for $u(x)$ all the functions given in the options one by one, but none seems to fit in. I substituted each $u(x)$ in the actual equation to see if the LHS equalled the RHS using WolframMathematica, but I could not figure out the correct option.
You are right. All answers seem incorrect! Try the general form $u(t)=e^{t}(at+b)$ and calculate $a$ and $b$ to satisfy the equation.
We get $$e^{x}(ax+b)+\dfrac{1}{2}(e^{x})\int_{0}^{1}(at+b)dt= 2xe^{x}.$$
After elementary calculations we get $a=2$, $b=\dfrac{-1}{3}$ and the function $$u(t)=e^{t}\bigg(2t-\dfrac{1}{3}\bigg)$$ indeed satisfies the equation