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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

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    $\begingroup$ Just to bring us back to the basics, we could ask the same thing about algebraic equations, or crazy equations with trig functions and whatnot. Or integral equations (single integrals, surface integrals with vector calculus involved). I think many ask this question. I usually like to wonder about existence first. Most graphs come from implicit equations. If you are comfortable knowing that implicit equations are important and often don't allow for explicit definition, then that's something we accept.This is why I made my existence comment. Just my 2 cents $\endgroup$
    – DWade64
    Sep 23, 2018 at 13:13
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    $\begingroup$ It does sound/become very philosophical doesn't it? Just because an equation exists, doesn't mean a solution does. That's why I find equations with no solution proofs interesting. $\endgroup$
    – Kendall
    Sep 23, 2018 at 13:18

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