Show HN: Differential Equations Explained
> Some of them don't even really have solutions (non-linear differential equations).<p>There certainly are differential equations without solutions, but I'm almost positive that's not what's meant here.  Rather, it is very likely that one can't <i>find</i> an explicit solution to a non-linear equation (EDIT: although there <i>are</i> non-linear equations that one <i>can</i> solve, such as $\dot y = 1/y$, whose solutions have graphs lying on straight lines through the origin); but there is a huge difference between an equation that doesn't <i>have</i> a solution and one whose solution we can't <i>find</i>.<p>For example, away from singularities, one can approximate the solution of a differential equation, even a non-linear one, numerically to an arbitrarily high degree of accuracy, almost exactly as the author does for linear equations.<p>EDIT:  TheLoneWolfling (<a href="https://news.ycombinator.com/item?id=9654767" rel="nofollow">https://news.ycombinator.com/item?id=9654767</a> )'s point that this numerical simulation can get hard very quickly is well taken, and probably more important in applications than abstract existence results.  Non-linear equations really are hard!