Expii

# 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.