Separable Differential Equation: Definition & How to Solve

Separable Differential Equation

A first order ordinary differential equation which can be solved by separating all occurrences of the two variables on either side of the equal sign and then integrating.

Example: $\frac{dy}{dx}=\frac{x^2}{y}$ Solution: $\begin{aligned} \frac{dy}{dx}&=\frac{x^2}{y} \\ y\,dy&=x^2\,dx \\ \int y\,dy&=\int x^2\,dx \\ \frac{1}{2}y^2&=\frac{1}{3}x^3+C \\ y^2&=\frac{2}{3}x^3+C \\ y&=\pm\sqrt{\frac{2}{3}x^3+C} \end{aligned}$

Key Formula

\frac{dy}{dx} = f(x)\,g(y) \quad\Longrightarrow\quad \frac{1}{g(y)}\,dy = f(x)\,dx \quad\Longrightarrow\quad \int \frac{1}{g(y)}\,dy = \int f(x)\,dx

Where:

$y$ = The dependent variable (unknown function of x)
$x$ = The independent variable
$f(x)$ = A function of x only
$g(y)$ = A function of y only
$\frac{dy}{dx}$ = The derivative of y with respect to x

Worked Example

Problem: Solve the differential equation dy/dx = 3x²y.

Step 1: Identify the separable form. The right side is the product of a function of x and a function of y.

\frac{dy}{dx} = 3x^2 \cdot y \quad\Rightarrow\quad f(x) = 3x^2,\; g(y) = y

Step 2: Separate the variables by dividing both sides by y and multiplying both sides by dx.

\frac{1}{y}\,dy = 3x^2\,dx

Step 3: Integrate both sides independently.

\int \frac{1}{y}\,dy = \int 3x^2\,dx \quad\Rightarrow\quad \ln|y| = x^3 + C

Step 4: Solve for y by exponentiating both sides.

|y| = e^{x^3 + C} = e^C \cdot e^{x^3} \quad\Rightarrow\quad y = Ae^{x^3}

Step 5: Here A = ±e^C is an arbitrary constant. Note that y = 0 is also a solution (set A = 0).

y = Ae^{x^3}, \quad A \in \mathbb{R}

Answer: The general solution is y = Ae^(x³), where A is an arbitrary constant.

Another Example

This example includes an initial condition (initial value problem) and shows a case where the solution is given implicitly or requires a cube root, unlike the first example where the solution was a clean exponential.

Problem: Solve the initial value problem dy/dx = (x + 1)/(y²) with y(0) = 2.

Step 1: Separate variables by multiplying both sides by y² and by dx.

y^2\,dy = (x + 1)\,dx

Step 2: Integrate both sides.

\int y^2\,dy = \int (x+1)\,dx \quad\Rightarrow\quad \frac{y^3}{3} = \frac{x^2}{2} + x + C

Step 3: Apply the initial condition y(0) = 2 to find C.

\frac{(2)^3}{3} = \frac{0}{2} + 0 + C \quad\Rightarrow\quad C = \frac{8}{3}

Step 4: Write the particular solution and solve for y.

\frac{y^3}{3} = \frac{x^2}{2} + x + \frac{8}{3} \quad\Rightarrow\quad y^3 = \frac{3x^2}{2} + 3x + 8 \quad\Rightarrow\quad y = \left(\frac{3x^2}{2} + 3x + 8\right)^{1/3}

Answer: y = ((3x²/2) + 3x + 8)^(1/3)

Frequently Asked Questions

How do you know if a differential equation is separable?

A first-order ODE is separable if you can write dy/dx as a product of a function of x alone and a function of y alone, i.e., dy/dx = f(x)·g(y). If the right side mixes x and y in a way that cannot be factored into such a product — for example, dy/dx = x + y — then the equation is not separable.

What is the difference between a separable and a linear differential equation?

A separable equation has the form dy/dx = f(x)·g(y), where variables can be split to opposite sides. A first-order linear equation has the form dy/dx + P(x)y = Q(x) and is solved using an integrating factor. Some equations are both separable and linear (like dy/dx = 2xy), while others are one but not the other.

Can you lose solutions when separating variables?

Yes. When you divide by g(y) to separate variables, any value of y where g(y) = 0 may be a constant solution (called an equilibrium or singular solution) that gets lost. You should always check whether g(y) = 0 produces a valid solution and state it separately if it does.

Separable Differential Equation vs. First-Order Linear Differential Equation

	Separable Differential Equation	First-Order Linear Differential Equation
Standard Form	dy/dx = f(x)·g(y)	dy/dx + P(x)y = Q(x)
Solution Method	Separate variables, then integrate both sides	Multiply by integrating factor μ(x) = e^(∫P(x)dx)
Can be nonlinear in y?	Yes (e.g., dy/dx = xy²)	No — y and dy/dx appear only to the first power
Overlap	Some separable equations are also linear	Some linear equations are also separable
Example	dy/dx = x/y	dy/dx + 2y = e^x

Why It Matters

Separable differential equations are typically the first type of ODE you learn to solve in a calculus or differential equations course, making them foundational. They model many real-world processes — exponential growth and decay, Newton's law of cooling, and simple population models are all separable. Mastering the separation-of-variables technique also builds the intuition needed for more advanced methods like integrating factors and substitution.

Common Mistakes

Mistake: Forgetting to check for lost solutions when dividing by g(y).

Correction: Before dividing by g(y), note any values where g(y) = 0. These may yield constant solutions (e.g., y = 0) that the general solution misses. Always check and list them separately.

Mistake: Forgetting the constant of integration or placing +C on both sides.

Correction: When you integrate both sides, each side technically gets its own constant, but you only need one. Combine them into a single constant C on one side of the equation. Omitting C entirely loses the family of solutions.

Related Terms

Differential Equation — General category that includes separable equations
Ordinary Differential Equation — Separable equations are a subtype of ODEs
First Order Differential Equation — Separable equations are always first order
Integration — Core operation used to solve separable equations
Variable — The two variables that are separated
Initial Value Problem — Uses an initial condition to find a particular solution