I don't like that attitude. There are many non-linear problems which can easily be solved accurately numerically or exactly.
So few people even try to solve problems exactly now that it seems like magic when someone does it. I can recall once instance where a colleague seemed awestruck that I solved Bernoulli's differential equation exactly when I noticed that a problem we were working on could be expressed that way: https://en.wikipedia.org/wiki/Bernoulli_differential_equatio...
If you do take this approach, I'd advise against explaining how you solved the problem. My colleague was not impressed that I looked up how to solve the problem, and thought what I did was "only" a trick. But there are many tricks for solving differential equations. Knowing which trick to pull out is part of my job as far as I'm concerned.
On that note, I'd recommend this website and the related books for finding exact solutions or useful changes of variables: http://eqworld.ipmnet.ru/
That's a good strategy, but you should have other tools for non-linear problems in your toolbox too.
I mostly want to push back against the idea that non-linear means unsolvable. My example transformed the problem to a linear one, but that's not the only possibility. Look at my other link for many other examples. Autonomous ordinary differential equations are one possible class of ODEs which don't need to be transformed to a linear system to be solved: https://en.wikipedia.org/wiki/Autonomous_differential_equati...
You can often solve a non-linear autonomous ODE by direct integration.
Worth noting that there's a large category of problems that don't seem linear, but can be approximated into a linear problem – like training a neural network!
