IE14

From Exampleproblems

Find the value of lambda for which the homogeneous Fredholm integral equation $y(x) = \lambda\int_0^1 e^xe^ty(t)dt\,$ has a nontrivial solution, and find all the solutions.

First, divide both sides by $\,e^x$ :
$\frac{y(x)}{e^x} = \lambda\int_0^1 e^t y(t)dt\,$
The right side is a constant; therefore, the left side must also be a constant. Thus $\frac{y(x)}{e^x} = C$ , so $\,y(x)= Ce^x$ .
Insert this in the integral equation: $Ce^x = \lambda\int_0^1 e^x e^t Ce^t\,dt$
Since we want a nontrivial solution, we may assume that $C \ne 0$ , so divide both sides by $\,Ce^x$ ; this yields $1 = \lambda\int_0^1 e^{2t}\,dt$
Evaluate the integral; this gives $1 = \lambda\frac{e^2 - 1}{2}$
Thus, a nontrivial solution exists only for $\lambda = \frac{2}{e^2-1}$ and the general solution in this case is $\,y(x) = Ce^x$ where $\,C$ is an arbitrary constant.