First order differential equations (HL only)

IB Mathematics: Analysis and Approaches HL· Unit 5: Calculus, Topic 12· 15 min read

1. Fundamentals and Separable Differential Equations★★☆☆☆HL only⏱ 5 min

📘 Definition

First order differential equation

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

An equation relating independent variable $x$ , dependent variable $y$ , and the first derivative of $y$ . The general solution includes an arbitrary constant; an initial condition fixes the constant for a particular solution.

Example:

\frac{dy}{dx} = 2xy

A separable ODE can be rearranged to group all $y$ terms on one side and all $x$ terms on the other, allowing direct integration of both sides.

📐 Worked Example

Find the general solution to $\frac{d y}{d x} = \frac{x ^{2}}{y ^{2}}$

1
Separate variables by multiplying both sides by $y^{2} d x$ :
$y^2 dy = x^2 dx$
2
Integrate both sides, add the constant of integration:
$\int y^2 dy = \int x^2 dx$
3
Evaluate the integrals:
$\frac{y^3}{3} = \frac{x^3}{3} + C$
4
Simplify to get the general solution:
$y = \sqrt[3]{x^3 + C}$

2. Homogeneous First Order Differential Equations★★★☆☆HL only⏱ 6 min

📘 Definition

Homogeneous first order ODE

A first order ODE of the form $\frac{d y}{d x} = f (\frac{y}{x})$ , solved using the substitution $v = \frac{y}{x}$ that transforms the ODE into a separable equation.

Since $y = v x$ , differentiate with respect to $x$ using the product rule to get $\frac{d y}{d x} = v + x \frac{d v}{d x}$ . Substitute this into the original ODE to get a separable equation in $v$ and $x$ .

📐 Worked Example

Solve the homogeneous ODE $x \frac{d y}{d x} = y + x$ , for $x > 0$

1
Rearrange to confirm it is homogeneous:
$\frac{dy}{dx} = \frac{y}{x} + 1 = f\left(\frac{y}{x}\right)$
2
Substitute $v = \frac{y}{x}$ and $\frac{d y}{d x} = v + x \frac{d v}{d x}$ :
$v + x\frac{dv}{dx} = v + 1$
3
Simplify and separate variables:
$dv = \frac{dx}{x}$
4
Integrate both sides:
$v = \ln x + C$
5
Substitute back $v = \frac{y}{x}$ for the general solution:
$y = x(\ln x + C)$

3. Linear First Order ODEs and Integrating Factor★★★★☆HL only⏱ 7 min

📘 Definition

Linear first order ODE

\frac{dy}{dx} + P(x)y = Q(x)

A first order ODE linear in $y$ , where $P (x)$ and $Q (x)$ are functions of $x$ only, solved using the integrating factor method.

The integrating factor is defined as $I (x) = e^{\int P (x) d x}$ . Multiplying through by $I (x)$ turns the left-hand side into the derivative of the product $I (x) y$ , which can be integrated directly:

\frac{d}{dx}\left(I(x)y\right) = I(x)Q(x)

📐 Worked Example

Find the general solution to $\frac{d y}{d x} + \frac{y}{x} = x^{2}$ , $x > 0$

1
Confirm the ODE is in standard linear form, with $P (x) = \frac{1}{x}$ and $Q (x) = x^{2}$ .
2
Calculate the integrating factor:
$I(x) = e^{\int \frac{1}{x} dx} = e^{\ln x} = x$
3
Multiply both sides by $I (x) = x$ :
$x\frac{dy}{dx} + y = x^3$
4
Recognize left-hand side as $\frac{d}{d x} (x y)$ :
$\frac{d}{dx}(xy) = x^3$
5
Integrate and rearrange for $y$ :
$xy = \frac{x^4}{4} + C \implies y = \frac{x^3}{4} + \frac{C}{x}$

4. Initial Value Problems in Applied Contexts★★★☆☆⏱ 4 min

In applied problems, you will almost always get an initial condition $y (x_{0}) = y_{0}$ , which lets you solve for the arbitrary constant $C$ and get a unique particular solution. Common applications include population growth, radioactive decay, Newton's law of cooling, and motion with resistance.

📐 Worked Example

A population grows according to $\frac{d P}{d t} = 0.02 P$ , with initial population $P (0) = 1000$ . Find $P$ when $t = 10$ .

1
Separate variables:
$\frac{dP}{P} = 0.02 dt$
2
Integrate both sides:
$\ln P = 0.02 t + C$
3
Exponentiate to simplify, rename $e^{C} = A$ :
$P = Ae^{0.02 t}$
4
Apply initial condition $P (0) = 1000$ :
$1000 = Ae^0 \implies A = 1000$
5
Calculate $P (10)$ :
$P(10) = 1000 e^{0.2} \approx 1221$

5. Common Pitfalls

Wrong move:

Forgetting to add the constant of integration when integrating both sides.

Why:

This gives an incorrect general solution, leading to wrong particular solutions and lost marks.

Correct move:

Add the constant of integration immediately after integrating.

Wrong move:

Calculating the integrating factor without writing the ODE in standard form with $d y / d x$ coefficient 1.

Why:

P(x) will be incorrect, leading to a wrong integrating factor and wrong solution.

Correct move:

Always divide through by the coefficient of $d y / d x$ first.

Wrong move:

Incorrect substitution for homogeneous ODEs, writing $d y / d x = d v / d x$ instead of $d y / d x = v + x d v / d x$ .

Why:

Forgetting the product rule for differentiating $y = v x$ gives a wrong separable equation.

Correct move:

Always use the product rule for the substitution.

Wrong move:

Dropping the absolute value when integrating $1/ x$ .

Why:

This leads to invalid solutions for negative $x$ , or incorrect integrating factors.

Correct move:

Keep the absolute value until simplifying, or confirm domain.

Wrong move:

Using integrating factor for non-linear ODEs.

Why:

The method only works for linear ODEs in $y$ , so it will not give a correct solution.

Correct move:

Classify the ODE first: separate, test homogeneity, then use integrating factor only for linear ODEs.

6. Quick Reference Cheatsheet

ODE Type	Standard Form	Solution Method
Separable	$\frac{d y}{d x} = f (x) g (y)$	$\int \frac{1}{g ( y )} d y = \int f (x) d x$
Homogeneous	$\frac{d y}{d x} = f (\frac{y}{x})$	Substitute $v = \frac{y}{x}$ , solve separable
Linear	$\frac{d y}{d x} + P (x) y = Q (x)$	Integrating factor $I (x) = e^{\int P (x) d x}$ , solve $\frac{d}{d x} (I (x) y) = I (x) Q (x)$
Initial Value	$\frac{d y}{d x} = f (x, y), y (x_{0}) = y_{0}$	Find general solution, substitute initial condition to find $C$

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.

2025 · 2
Solve linear ODE with initial condition
2023 · 1
Solve homogeneous first order ODE
2022 · 2
Separable ODE population model

What's Next

First order differential equations are the foundation for more advanced topics in calculus and dynamic systems modeling, widely used across STEM fields. Mastering classification of ODE types and their solution methods is critical for IB AA HL exams, where mixed problems requiring you to select the right method are common. Next, you will extend your knowledge to second order ODEs, used to model oscillations, motion, and electric circuits. You will also learn qualitative analysis of ODEs with slope fields and equilibrium solutions, which helps check your results and interpret solutions in context. Practicing mixed problems that require classification first is the best way to prepare for exam questions.