§7.4 Separable differential equations

Introduction

Among the differential equations $\frac{dy}{dt} = -3y$, $\frac{dy}{dt} = \frac{y}{2-t}$, and $\frac{dy}{dt} = t^2-y^2$, the one that is not separable is

To separate $\frac{dy}{dt} = -3y$, we divide both sides by $y$ and obtain

For $\frac{dy}{dt} = t\,y$, proper separation followed by integration gives

The equation $\frac{dy}{dt} = ty-y$ separates as

The equation $\frac{dy}{dt} = -3y$ has solutions $y = Ce^{-3t}$. For the solution with $y(0) = 6$, find the constant $C$.

The equation $\frac{dy}{dt} = t+1$ has solutions $y = \frac{t^2}{2}+t+C$. For the solution with $y(0) = 4$, find the constant $C$.

An equation of the form $\frac{dy}{dt} = g(t)$, such as $\frac{dy}{dt} = t+1$, counts as separable since the right side is already a pure function of $t$.