10

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.

ViktorStein
  • 4,838
  • 2
    Can this be reduced to a question about visual intuition for the chain rule? – Lee Mosher Jul 30 '19 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. – ViktorStein Jul 30 '19 at 23:57
  • I think this link will help you https://math.stackexchange.com/questions/1525791/differential-equations-how-does-separation-of-variables-really-work – nmasanta Jul 31 '19 at 07:48

1 Answers1

1

Separation of variables are quite useful in solution of DE's in physics for the applicability physical system for instance harmonically vibrating membrane or wave equation of a particle (there will be several constraints to apply). Wave equation of these systems can generally be written as separate variables of time and position, so to solve the DE to get the governing equation of physical system's , dependence relation to its variables allows to independent terms to be written separately in terms of independent variables. So, for the mathematical demonstration of this relation is basically related to integrals or derivatives of composite functions (as @Lee Mosher said in the comment) and multiplication of independent variable functions ( f(g(x)) and f(x)g(y) ).

asd.123
  • 331