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⬅ Back to Differential Equations
📘 Key Concepts: Separable Equations

A separable differential equation can be rewritten so that all $y$-terms are on one side and all $x$-terms are on the other:

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

Steps:

  1. Separate variables: move all $y$-terms (with $dy$) to the left, all $x$-terms (with $dx$) to the right.
  2. Integrate both sides.
  3. Solve for $y$ (explicit or implicit), then apply initial conditions to find $C$.

Common models:

Problem 1: Basic Separable Equation

Solve the differential equation $\dfrac{dy}{dx} = 2xy$ given the initial condition $y(0) = 3$.

DE – Separable – Problem 1 – Setup DE – Separable – Problem 1 – Diagram DE – Separable – Problem 1 – Diagram
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Problem 2: Trigonometric Separable Equation

Solve $\dfrac{dy}{dx} = y\cos x$ given $y(0) = e$.

DE – Separable – Problem 2 – Setup DE – Separable – Problem 2 – Diagram DE – Separable – Problem 2 – Diagram
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Problem 3: Population Growth

A bacterial colony numbers 500 at $t = 0$ and grows to 800 at $t = 2$ hours. Assuming exponential growth governed by $\dfrac{dP}{dt} = kP$:
a. Find the growth constant $k$.
b. Predict the population at $t = 5$ hours.

DE – Separable – Problem 3 – Setup DE – Separable – Problem 3 – Diagram DE – Separable – Problem 3 – Diagram
Solution Solution Solution Solution

Problem 4: Implicit Solution

Solve $(1 + x^2)\,dy = xy\,dx$ given $y(0) = 2$. Express the solution explicitly for $y$.

DE – Separable – Problem 4 – Setup DE – Separable – Problem 4 – Diagram DE – Separable – Problem 4 – Diagram
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