If $\phi(x)=\lambda\int_0 ^1 e^{x+t}\phi(t)dt$, then for what value of $\lambda$ does there exist a non-trivial solution?

Question

For the homogeneous Fredholm integral equation $$\phi(x)=\lambda\int_0 ^1 e^{x+t}\phi(t)dt$$ For what value of $\lambda$ does there exist a non-trivial solution for $\phi(x)$?

Answer 1

$$\frac{\phi(x)}{e^x}=\lambda\int_0^1 e^t\phi(t)dt$$

The RHS is a constant, let it be $c$, so we have

$$\frac{\phi(x)}{e^x}=\lambda\int_0^1 e^t\phi(t)dt=c$$

$$\Rightarrow \phi(x)=c\cdot e^x$$

$c\neq 0$ otherwise the solution is trivial. Plug into the initial equation:

$$c=\lambda\int_0^1e^t\cdot c\cdot e^t dt$$

$$\Rightarrow\lambda=\frac{2}{e^2-1}$$