TITLE: Why is there no general solution for the general 2nd order linear ODE
QUESTION [6 upvotes]: We can always solve a general first order linear ODE:
$$y'(x)+a(x)y(x)=b(x).$$
I am looking for some intuition why the general 2nd order linear ODE
$$y''(x)+a(x)y'(x)+b(x)y(x)=c(x) $$
does not have a gerneral formula. Is it mathematically impossible, or is there a chance, that someone will find a general solution? If it is mathematically impossible, is there any intuitive explanation to this phenomenon?

REPLY [2 votes]: It is mathematically impossible. The proof goes (generally speaking) along the same lines as it is proved that there is no general formula to solve algebraic equations of degree 5 or up. All the details can be found in, e.g., An introduction to differential algebra 
by Irving Kaplansky.