First put to use the fact that both v and w satisfy the same differential equation as y:
Now look at the dif. eq. in terms of y
. If is the solution to the equation than using such y should lead to a mathematical identity. First notice that and and use that in the equation:
Now use the expressions for v'' and w''.
Everything on the left side cancels out and you have the identity which is of course true, thus proving that the solution indeed can be defined as y=v+w with v and w satisfying the same diff. eq. as y.
Not sure, but maybe it will help you somehow.
Anyway, v and w here are solutions to the diff. eq. and their sum is also a solution.