So this questions relates to my specific ODE but also ODEs in general.
I am a big fan of solving ODEs by hand, but I also know when to give up and use, say, Mathematica to solve it for me. Having said that, a lot of ODEs including nth-order linear ODEs such as an Euler-Cauchy type equation are often solvable by hand and aren't too lengthy. However, ODEs such as my 2nd-order nonlinear ODE
$$xf(x)+af'(x) [f'(x)^2+bf''(x)^2)]^{1/2}=0$$
are not easily identified if DEs aren't your speciality.
So my question is this, how does one go about deciding if a DE is solvable by hand, i.e. could I solve the above ODE by hand? If it is solvable by hand, how do we "know"/"decide" which method to use if it isn't obvious? Lastly, if we resort to using software and it fails to solve it analytically, does that necessarily imply that only a numerical solution exists?
Many thanks for all the help and feedback
Ken