# Is there any visual intuition for separation of variables?

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.

• Can this be reduced to a question about visual intuition for the chain rule? – Lee Mosher Jul 30 at 23:27
• @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. – Viktor Glombik Jul 30 at 23:57
• I think the integral on the right should go from $t_0$ to $t$ (instead of $u(t_0)$ to $u(t)$). Right? – iljusch Aug 6 at 19:37