Solving Separable Differential Equations - Expii

A separable differential equation is an equation of the form \[\frac{dy}{dx} = g(x)h(y),\] where \(g,h\) are given functions of \(x,y\), respectively. For example, when \(h\) is constant, we recover the anti-derivative equation. In general, you can solve separable equations essentially by "separating the two variables \(x\) and \(y\)": \[ \frac{dy}{h(y)} = g(x) dx \implies \int\frac{dy}{h(y)} = \int g(x) dx \implies H(y) = G(x) + C, \] where \(G\) and \(H\) are anti-derivatives of \(g\) and \(1/h\), respectively, and \(C\) is some constant depending on the initial conditions. Alternatively, we could have solved the equation using the chain rule: \[g(x) = \frac{1}{h(y)}\frac{d}{dx}y(x) = \frac{d}{dx} H(y(x)) \implies H(y(x)) = G(x) + C.\] Either way, you can then usually solve for \(y(x)\) in terms of \(x\) by applying the inverse function \(H^{-1}\) to both sides.