Separable Differential Equations

Separable equations are the simplest and most common type of first-order differential equation. An equation is separable when you can rearrange it so that all terms involving $y$ (and $dy$) are on one side and all terms involving $x$ (and $dx$) are on the other. Once separated, you simply integrate each side independently — a technique that reduces a differential equation to two ordinary integration problems you already know how to handle.

The classic example is the exponential growth/decay equation $y' = ky$, which models everything from bacterial populations to radioactive isotopes to continuously compounded interest. Separating gives $\frac{1}{y}\,dy = k\,dx$; integrating both sides yields $\ln|y| = kx + C$, or equivalently $y = Ce^{kx}$. When $k > 0$ the solution grows without bound; when $k < 0$ it decays toward zero. An initial condition like $y(0) = y_0$ pins down the constant $C = y_0$.

Not every first-order equation is separable — equations like $y' = x + y$ have the two variables tangled in a way that cannot be split. Recognizing separability quickly is an important skill, because when it works the solution method is fast and mechanical. In this lesson you will practice the separation-and-integration technique on several families of equations and use the interactive explorer to see how the growth constant $k$ and initial value $y_0$ shape the exponential solution curve.