# IE14

Find the value of lambda for which the homogeneous Fredholm integral equation $y(x)=\lambda \int _{0}^{1}e^{x}e^{t}y(t)dt\,$ 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 $C\neq 0$, 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.