This meets your requirements.

If you want a general solution, you need a solution of y''(t)-ty'(t)+y(t)=0.

The derivation on both sides of the equation is [y''(t)-ty'(t)+y(t)]'=0.

That is y'' (t) = ty'' (t)

So you can get y'' (t) = c [2] e (t/2).

This shows that the general solution is not an elementary function.

If you want to find a general solution, just expand E (t/2) and divide it by 2.

Of course, then you have to make it meet your initial conditions.