In this post Solving a separable 2nd order differential equation (can a similar technique be used)? we can see a way to solve the following differential equation: $y''=-\frac{k}{y^2}$. And the author uses the idea of division of derivatives in Leibniz notation and then manipulating with differentials as numbers. My question is, how can such a manipulation be rigorously justified ?
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If you have $y''=f(y)$, you multiply with $y'$ and integrate to $$ \frac12y'^2=\int f(y) dy $$ applying the usual substitution rules.
The second approach uses some implicit function argument. You can posit that as long as the solution is not constant, you can parametrize it, at least locally, by $y$. In consequence, there is some function $v$ so that $y'=v(y)$. Taking the derivative, $y''=v'(y)y'=v'(y)v(y)$ leading to $$\int v\,dv=\int f(y)\,dy.$$
Lutz Lehmann
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