I hate to see numerical approximation methods used when there exists a very simple and precise method done by hand. First we designate by K the integral of t*y(t) from 0 to 1, which is unknown as yet. This gives
Integrating this w.r. to x gives
y(x) = x + (K-1/3)*x^2/2 + C
where C is the unknown constant of integration. However, since y(0) = 0, this implies that C = 0. Now we have
t*y(t) = t^2 + (K-1/3)*t^3/2
Integrating t*y(t) from 0 to 1 gives t^3/3 + (K-1/3)*t^4/8 evaluated at t = 1 minus its value at t = 0, so that gives
which has the unique solution K = 1/3. This in turn gives us our final answer:
No need for matlab or numerical approximations.