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Separation of variables is a standard procedure to solve a differential equation of the form $$ u'(t) = g(t) h(u(t)) $$ transforming it to via division and substitution to $$ \int_{u(t_0)}^{u(t)} \frac{ds}{h(s)} = \int_{t_0}^{t} g(s) ds. $$

All algebraic manipulations make perfect sense to me but I wondered if there is any visual intuition to why this approach works.

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    $\begingroup$ Can this be reduced to a question about visual intuition for the chain rule? $\endgroup$ – Lee Mosher Jul 30 at 23:27
  • $\begingroup$ @LeeMosher Even though this is an interesting approach I hadn't thought of, I think I'd prefer something more ODE-specific, maybe working with stream plots or another way to visualise a first differential equation. $\endgroup$ – Viktor Glombik Jul 30 at 23:57
  • $\begingroup$ I think this link will help you math.stackexchange.com/questions/1525791/… $\endgroup$ – nmasanta Jul 31 at 7:48
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    $\begingroup$ I think the integral on the right should go from $t_0$ to $t$ (instead of $u(t_0)$ to $u(t)$). Right? $\endgroup$ – iljusch Aug 6 at 19:37
  • $\begingroup$ @iljusch You're right, I corrected it. $\endgroup$ – Viktor Glombik Aug 7 at 6:05

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