Separable Equations

You will master the technique of solving separable differential equations by separating variables, integrating both sides, and solving for the function. This is one of the most useful and widely applicable solution methods for first-order equations.

Many first-order differential equations can be written in the form:

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

When an equation can be arranged like this, we say it is separable. The strategy is simple but powerful: move all the $y$ terms to one side and all the $x$ terms to the other, then integrate both sides.

Starting from:

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

We rewrite it as:

$$\frac{1}{g(y)} \, dy = f(x) \, dx$$

Then integrate both sides:

$$\int \frac{1}{g(y)} \, dy = \int f(x) \, dx + C$$

This gives us an implicit solution that we can often solve explicitly for $y$.

The visual above shows the family of solution curves for $\frac{dy}{dx} = xy$. Each curve corresponds to a different value of $C$ in the general solution $y = Ce^{x^2/2}$.

Example 1: Basic Separation

Solve $\frac{dy}{dx} = \frac{x}{y}$.

Solution:

$$y \, dy = x \, dx$$

Integrate both sides:

$$\int y \, dy = \int x \, dx$$ $$\frac{y^2}{2} = \frac{x^2}{2} + C$$

Multiply through by 2:

$$y^2 = x^2 + K \quad \text{where } K = 2C$$

Example 2: With Initial Condition

Solve $\frac{dy}{dx} = 2xy$ with $y(0) = 3$.

Solution:

$$\frac{dy}{y} = 2x \, dx$$ $$\ln |y| = x^2 + C$$ $$y = Ae^{x^2} \quad \text{(where } A = \pm e^C\text{)}$$

Apply $y(0) = 3$: $3 = A \cdot e^0 \implies A = 3$

Final solution: $y = 3e^{x^2}$

Example 3: Logistic Growth

Solve the logistic equation $\frac{dP}{dt} = 0.2P\left(1 - \frac{P}{100}\right)$ with $P(0) = 20$.

Solution: Separate variables and use partial fractions (details in later lessons), leading to:

$$P(t) = \frac{100}{1 + 4e^{-0.2t}}$$

Separable equations are used constantly in biology and chemistry. The logistic growth model you saw above predicts how populations grow when resources are limited. Pharmacologists use separable equations to model how drug concentration in the bloodstream decreases over time (exponential decay). Environmental scientists use them to track pollutant levels in lakes or the spread of invasive species. Even your phone’s battery drain over time can be modeled with a simple separable equation.