I'm working through an example in my notes:

Suppose y = y(x) satisfies the integral equation

..............(1)

Show it implies a boundary value problem for y.

Splitting integral at x = t:

..............(2)

The example then says by eliminating the integral term between equations 1 and 2 implies the differential equation y'' - 15y = -9x

I can't see where this has come from at all, could someone please explain? Which integral term??

Thanks in advance!