You an immediately translate this integral equation into an easy second order linear ODE on $[0,1]$ with null boundary conditions (I'm not going to do it for you). Just note that your integrand $ts-\min(t,s)$ is the [Green function][1] of the Laplacian in dimension $1$ (i.e. the second derivative) with Dirichlet boundary condition: one has
$$\int_0^1\big(sr-\min(t,s)\big)x(t)ds=u(t)$$
if and only  $$\cases{\ddot u(t)=x(t)\\u(0)=0\\u(1)=0\,.}$$
(The easy computation start writing $\int_0^1=\int_0^t+\int_t^1$. Then you may want to precise further the above claim in the various functional setting; however for your eigenvalue problem everything is necessarily classic and smooth). 


[1]:https://en.wikipedia.org/wiki/Green%27s_function