Separable Equations

A separable differential equation is a first-order ODE that can be written in the form where all terms involving the dependent variable appear on one side and all terms involving the independent variable appear on the other side.

DefinitionSeparable Equation

A first-order differential equation is called separable if it can be written in the form:

$\frac{dy}{dx} = g(x)h(y)$

or equivalently, after separation:

$\frac{dy}{h(y)} = g(x)dx$

The solution method involves separating variables and integrating both sides. This technique is one of the most fundamental approaches to solving first-order ODEs and applies to a wide class of problems in physics, biology, and engineering.

Solution Method

To solve a separable equation $\frac{dy}{dx} = g(x)h(y)$ :

Separate variables: Write the equation as $\frac{dy}{h(y)} = g(x)dx$
Integrate both sides: $\int \frac{dy}{h(y)} = \int g(x)dx$
Solve for y: If possible, solve the resulting equation explicitly for $y$

ExamplePopulation Growth

Consider the differential equation $\frac{dy}{dt} = ky$ where $k > 0$ is a constant (exponential growth model).

Solution: This is separable with $g(t) = k$ and $h(y) = y$ .

$\frac{dy}{y} = k\,dt$

Integrating both sides:

$\ln|y| = kt + C$

Solving for $y$ :

$y = Ae^{kt}$

where $A = \pm e^C$ is determined by the initial condition. This is the classic exponential growth equation describing population dynamics, radioactive decay, and compound interest.

Remark

When separating variables, we must be careful about dividing by functions that could be zero. If $h(y_0) = 0$ for some value $y_0$ , then $y = y_0$ is also a solution (called an equilibrium solution). These constant solutions must be checked separately.

The method of separation of variables is powerful because it reduces a differential equation to two independent integration problems. However, the resulting integrals may not always have closed-form solutions, and even when they do, solving explicitly for $y$ may be difficult or impossible.

ExampleLogistic Equation

The logistic equation $\frac{dy}{dt} = ry(1 - y/K)$ models population growth with carrying capacity $K$ .

This is separable:

$\frac{dy}{y(1-y/K)} = r\,dt$

Using partial fractions on the left side:

$\frac{1}{y(1-y/K)} = \frac{1}{y} + \frac{1/K}{1-y/K}$

Integration yields:

$\ln|y| - \ln|1-y/K| = rt + C$

Solving gives the logistic solution:

$y(t) = \frac{K}{1 + Ae^{-rt}}$

where $A$ depends on initial conditions. This S-shaped curve is fundamental in ecology and epidemiology.