Concept preview

What you will be able to do

1. 01Recognise a separable differential equation $\dfrac{dy}{dx}=g(x)\,h(y)$ and rewrite it as $\dfrac{dy}{h(y)}=g(x)\,dx$ with all $y$-material on one side and all $x$-material on the other.
2. 02Integrate both sides, introduce a single constant of integration $+C$, and solve algebraically for $y$ to obtain the general solution (a family of curves).
3. 03Apply separation to the proportional-rate model $\dfrac{dy}{dx}=ky$ to derive $y=Ce^{kx}$, and explain why the constant of integration must be kept until after integrating.

A differential equation is separable if you can write it as $\dfrac{dy}{dx}=g(x)\,h(y)$ — the slope factors into an $x$-only piece times a $y$-only piece. When it does, treat $\dfrac{dy}{dx}$ as a ratio of differentials (a move that is justified rigorously by the chain rule, but which behaves exactly like ordinary algebra here) and gather like with like. Divide both sides by $h(y)$ and multiply both sides by $dx$: $\dfrac{dy}{h(y)}=g(x)\,dx.$ Now every $y$, including the $dy$, sits on the left, and every $x$, including the $dx$, sits on the right. The variables are separated.