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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.

Calculus : Integral Equations