Separation of variables for differential equations

IB Mathematics AI HL· 25 min read

1. Identifying Separable Differential Equations★★☆☆☆⏱ 8 min

A first-order differential equation is separable if it can be rearranged to isolate all $y$ terms on one side and all $x$ terms on the other.

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

📘 Definition

Separable differential equation

A first-order differential equation that can be split into two independent functions of $x$ and $y$ , allowing separate integration of each variable.

Example:

$\frac{d y}{d x} = x y$ is separable, while $\frac{d y}{d x} = x + y$ is not.

📐 Worked Example

Determine if $\frac{d y}{d x} = \frac{x y ^{3}}{1 + x ^{2}}$ is separable.

1
Factor the right-hand side into products of functions of $x$ only and $y$ only:
2
$\frac{dy}{dx} = \left(\frac{x}{1 + x^2}\right) \times y^3$
3
Since the first term depends only on $x$ and the second only on $y$ , the equation is separable.

✓ Quick check

Confirm your understanding:

Is $\frac{d y}{d x} = y^{2} x$ separable?
- Yes
- No
Is $\frac{d y}{d x} = x + 2 y$ separable?
- Yes
- No
Reveal answer
1 —
This cannot be split into a product of $x$ -only and $y$ -only functions, so it is not separable.

2. Finding the General Solution★★★☆☆⏱ 10 min

Once you confirm an equation is separable, follow these three core steps to get the general solution, which includes the arbitrary constant of integration representing the full family of solutions.

Rearrange to group all $y$ terms with $d y$ on one side, and all $x$ terms with $d x$ on the other
Integrate both sides with respect to their respective variables
Simplify to express $y$ as a function of $x$ , leaving the constant of integration in place

📐 Worked Example

Find the general solution of $\frac{d y}{d x} = 3 x^{2} y$ .

1
Separate variables:
2
$\frac{1}{y} dy = 3x^2 dx$
3
Integrate both sides:
4
$\int \frac{1}{y} dy = \int 3x^2 dx$
5
Evaluate integrals and add the constant of integration:
6
$\ln|y| = x^3 + C$
7
Exponentiate to solve for $y$ , where $A = e^{C}$ is an arbitrary constant:
8
$y = A e^{x^3}$

3. Finding Particular Solutions with Initial Conditions★★★☆☆⏱ 7 min

Most exam problems give an initial condition (a point $(x_{0}, y_{0})$ that the solution passes through) to find a single specific solution, called a particular solution.

📘 Definition

Particular solution

A specific solution that satisfies both the differential equation and the given initial condition, with no arbitrary constants remaining.

📐 Worked Example

Given $\frac{d y}{d x} = 3 x^{2} y$ with $y (0) = 2$ , find the particular solution.

1
Start from the general solution found earlier:
2
$y = A e^{x^3}$
3
Substitute the initial condition $x = 0, y = 2$ to solve for $A$ :
4
$2 = A e^{0^3} = A(1) \implies A = 2$
5
Substitute $A$ back to get the particular solution:
6
$y = 2 e^{x^3}$

4. Applied Problems in Context★★★★☆HL only⏱ 10 min

Separation of variables is widely used to solve real-world rate problems, including population growth, radioactive decay, and cooling. You will first translate the problem description into a differential equation, then solve it using the steps above.

📐 Worked Example

The rate of growth of a bacteria population is proportional to the current population $P$ at time $t$ hours. At $t = 0$ , the population is 1000. After 2 hours, it is 3000. Find $P (t)$ .

1
Translate the problem into a differential equation, where $k$ is a constant of proportionality:
2
$\frac{dP}{dt} = kP$
3
Separate and integrate to get the general solution:
4
$\int \frac{1}{P} dP = \int k dt \implies P(t) = A e^{kt}$
5
Use the initial condition $P (0) = 1000$ to find $A$ :
6
$1000 = A e^{0} \implies A = 1000$
7
Use the second condition $P (2) = 3000$ to find $k$ :
8
$3000 = 1000 e^{2k} \implies 2k = \ln 3 \implies k = \frac{1}{2}\ln 3$
9
Write the final simplified solution:
10
$P(t) = 1000 (3)^{\frac{t}{2}}$

5. Common Pitfalls

Wrong move:

Forgetting to add the constant of integration after integrating

Why:

Without the constant, you only get one arbitrary solution, not the general solution required for full marks

Correct move:

Add the constant of integration to one side immediately after integrating both sides

Wrong move:

Trying to separate non-separable differential equations

Why:

You will waste time in the exam and cannot get a correct solution with this method

Correct move:

Check if the right-hand side can be factored into $f (x) g (y)$ before starting separation

Wrong move:

Adding separate constants to both the $x$ and $y$ sides

Why:

This creates unnecessary extra work and increases the risk of algebraic errors

Correct move:

Only add one constant to the $x$ side after integrating both sides

Wrong move:

Substituting the initial condition before integrating

Why:

You lose the information needed to correctly calculate the constant of integration

Correct move:

Find the full general solution first, then substitute the initial condition to find the constant

6. Quick Reference Cheatsheet

Step	Action
1	Check if $\frac{d y}{d x} = f (x) g (y)$ (separable)
2	Rearrange: $\frac{1}{g ( y )} d y = f (x) d x$
3	Integrate both sides, add $C$ to one side
4	Simplify to get general solution
5	Substitute initial condition for particular solution

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2021 · 2
8-mark initial value problem
2023 · 1
5-mark population growth problem

What's Next

Separation of variables is the foundation for all differential equation work in IB Math AI HL, and is regularly tested in both paper 1 and paper 2, often in applied context problems worth 5-8 marks. Mastering this method will prepare you for numerical methods for differential equations and more complex population models, which are common topics in the final exam.