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)?

https://groups.google.com/group/sci.math/topics

or (which is the same group)

https://mathforum.org/kb/forum.jspa?forumID=13