Transform the integral equation into a Sturm-Liouville problem using eigenfunction expansions Transform the integral equation $\int_0^1 k(x,y)u(y)dy-\lambda u(x)=x$ into a Sturm-Liouville problem using eigenfunction expansions, where $k(x,y)=x(1-y)$ if $x<y$, and $k(x,y)=y(1-x)$ if $x>y$.

You can recognise the function $k(x,y)$ as the Green's function for the operator $L = \frac{\text{d}^2}{\text{d} x^2}$ on the domain $[0,1]$ with homogeneous Dirichlet boundary conditions. If you now would be given the inhomogeneous differential equation $L v = f$, then the solution to that equation would be given as \begin{equation} v(x) = \int_0^1 k(x,y) f(y)\,\text{d} y. \end{equation} To make this equal to the integral equation from your question, we have to choose $v(x) = \lambda u(x) + x$ and $f(x) = u(x)$. That means that the original ODE $L v = f$ takes the form $v'' = u$. So, combining these observations, you obtain the ODE for $v$ \begin{equation} \lambda v'' - v = -x. \end{equation} Alternatively, you can infer from the integral equation that the boundary conditions for $u$ are given by $\lambda u(0) = 0$ and $\lambda u(1) = -1$; you can then express $u$ in terms of the eigenfunctions of $L$ on the domain $[0,1]$ with these boundary conditions, and see what you get.