The equation

du/dy = R(r) - 1/2 u²....................(1)

belongs to a very useful class of equations called Riccati equations. The case where R(t) is a non-negative function plays a very important role in geometry, cosmology, awl Einstein's general theory of relativity. 

(a) Say that we know a solution u(t) of this equation. We claim that then w(t) = u(t) - 1/v(t) is another solution.3 Show that this claim is true if v(t) is itself a solution of a certain linear differential equation, which you should find. 

(b) If R(t) = 1/2(1+t)², then u(t) = 1/1+t a solution of equation (1) with initial data u(0) = 1. Find v(t), t ≥ 0, in this case and use it to find a solution w(t) of equation (1) with initial data w(0) = 2.