Find the value of lambda for which the homogeneous Fredholm integral equation has a nontrivial solution, and find all the solutions.
First, divide both sides by :
The right side is a constant; therefore, the left side must also be a constant. Thus , so .
Insert this in the integral equation:
Since we want a nontrivial solution, we may assume that , so divide both sides by ; this yields
Evaluate the integral; this gives
Thus, a nontrivial solution exists only for and the general solution in this case is where is an arbitrary constant.