One of my advisors is asking me if I know whether such a differential equation has a closed form solution. y′′(t)(ty′(t)-y(t)+k)=y(t)

for t in

where y(a)=0 and y'(a)=r

The function y(t)=(1/2)t^2 + 1/2 k is definitely a solution but it doesn't necessarily satisfy the initial conditions.

Is this a well known equation type? Is it known to have a closed form solution (or not)?