What is the solution of this Fredholm integral equation?

Question

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.

Why would someone vote to close this question? – F. A. Mala Jul 28 '22 at 16:34 — F. A. Mala, Jul 28 '22 at 16:34

score 3 · Accepted Answer · edited Jul 28 '22 at 17:19

3

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

edited Jul 28 '22 at 17:19

F. A. Mala

1

answered Jul 28 '22 at 17:07

Thanks for the edit! An upvote would be highly appreciated! – Jul 28 '22 at 17:16

What is the solution of this Fredholm integral equation?

1 Answers1