I have the following ODE: ($y:[0,1]\rightarrow \mathbb{R} , t\mapsto y(t)$ a function) $$y''(t) -t y'(t) -2 y(t) = 0, y(0)=1, y'(0)=a\in \mathbb{R}.$$
Because there is a factor $t$ in front of $y'(t)$ this equation does not have constant coefficients. I found a recipe here: http://www.math.psu.edu/tseng/class/Math251/Notes-2nd%20order%20ODE%20pt1.pdf of how to solve 2nd order ODE's but this case seems to be difficult.
I can't find any particular solution to this equation. Does anyone know how to solve it or proceed? Any hint is highly appreciated.
Thanks!