Topic 5.24 · HL only

Linear first order differential equations: notes and practice questions

Summary

The Maclaurin series is a Taylor series expansion of a function $f(x)$ centered at $a=0$ .
It expresses $f(x)$ as an infinite sum of terms involving its derivatives evaluated at $x=0$ .
The general formula is $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n$ .
The Maclaurin polynomial of degree $n$ , $M_n(x)$ , is the sum of the first $n+1$ terms of the series.

How it is examined

Four methods under one code, and the question usually names the method, so generation should name it too. The logistic equation is the IB's own example and it pulls in partial fractions, which makes it a good multi-part question. Euler's method is arithmetic that has to be laid out in a table, and it is a Paper 2 item. The constant of integration must be found from the initial condition before rearranging, not after. 8 to 12 marks across parts.

Given in the booklet

Euler's method as $y_{n+1} = y_n + h \times f(x_n, y_n)$ ; $x_{n+1} = x_n + h$ , where $h$ is a constant (step length), and the integrating factor $e^{\int P(x)\mathrm{d}x}$ for $y' + P(x)y = Q(x)$ . **The homogeneous substitution $y = vx$ is not in the booklet**, it is in the syllabus guidance and has to be recalled.

Key ideas

First order differential equations.
Numerical solution of $\dfrac{\mathrm{d}y}{\mathrm{d}x} = f(x, y)$ using Euler's method.
Variables separable.
Homogeneous differential equation $\dfrac{\mathrm{d}y}{\mathrm{d}x} = f\left(\dfrac{y}{x}\right)$ using the substitution $y = vx$ .

Not assessed

First order only. Second order differential equations are not on the syllabus.

Linking questions

Other contexts: Newton's law of cooling, population growth, carbon dating.
Links to other subjects: decay curves (physics); first order reactions (chemistry).

Practice questions

7 questions · 3 medium · 4 hard

Showing 7 of 7

Question 1

MediumPaper 1 · no calculator7 marks

Solve the differential equation $x\frac{dy}{dx} + y = x \cos x$ , for $x > 0$ , given that $y = 3$ when $x = \pi$ .

Give your answer in the form $y = f(x)$ .

Question 2

HardPaper 1 · no calculator7 marks

In this question you may assume the result $\int \sec x \mathrm{d}x = \ln(\sec x + \tan x) + c$ .

Consider the differential equation $\cos x \frac{\mathrm{d}y}{\mathrm{d}x} - 2y \sin x = \sec^2 x$ for $0 \le x < \frac{\pi}{2}$ .

Given that $y = 5$ when $x = \frac{\pi}{4}$ , find the particular solution for this differential equation.

Express your answer in the form $y = f(x)$ .

Question 3

MediumPaper 2 · calculator10 marks

(a)

A metal object is cooling in a room. Its temperature, $T$ (in degrees Celsius), at time $t$ (in minutes) can be modelled by the differential equation $\frac{dT}{dt} = -0.05(T-20)$ . Initially, at $t=0$ , the temperature of the object is $80^{\circ}\text{C}$ .

(a) Use Euler's method with a step size of 0.1 to find an approximation for the temperature of the object when $t=0.4$ minutes. Give your answer correct to four significant figures.

[3]

(b)

(b) By solving the differential equation, show that $T = 20 + 60e^{-0.05t}$ .

[5]

(c)

[2]

Question 4

HardPaper 1 · no calculator38 marks

(a)

Find the general solution to the following differential equation. (a)

$\frac{dy}{dx} = \frac{y^2}{x^2-1}$

[5]

(b)

(b)

$\frac{dy}{dx} = y^2 \sin x \cos x$

[5]

(c)

(c) Find the particular solution to the differential equation $(x^2+4) \frac{dy}{dx} = xy$ , given the initial condition $y(0)=1$ .

[6]

(d)

(d)

$e^x \frac{dy}{dx} = \frac{1}{y}$

[4]

(e)

(e)

$\frac{dy}{dx} + 2y = xe^{-x}$

[6]

(f)

(f) $x \frac{dy}{dx} - 3y = x^5$ for $x>0$ .

[6]

(g)

(g) $\frac{dy}{dx} + y \cot x = \cos x$ for $0 < x < \pi$ .

[6]

Question 5

MediumPaper 2 · calculator7 marks

The population $P$ (in thousands) of a certain bacterial culture at time $t$ (in hours) is modelled by the differential equation $\frac{dP}{dt} + 2P = t$ , for $t \ge 0$ . Initially, at $t=0$ , the population of the bacterial culture is $1$ thousand.

Solve this differential equation to find $P$ as a function of $t$ .

Question 6

HardPaper 2 · calculator15 marks

(a)

The rate of change of the concentration, $C$ , of a reactant in a chemical process at time $t$ is modelled by the differential equation $\frac{dC}{dt} = 2t - \frac{C}{t}$ , for $t > 0$ . Given that the initial concentration is $C(1) = 2$ , use Euler's method with a step size of $h = 0.2$ to find an approximate value of $C$ when $t = 1.6$ .

[4]

(b)

Solve the differential equation $\frac{dC}{dt} = 2t - \frac{C}{t}$ using an appropriate analytical method, and find the exact value of $C$ when $t = 1.6$ .

[7]

(c)

Sketch the approximate values you found in part (a) and the graph of the function you found in part (b) on the same coordinate axes. Use your sketch to explain why your answer in part (a) is greater than or less than the value in part (b).

[4]

Question 7

HardPaper 1 · no calculator7 marks

Solve the differential equation $x\frac{dy}{dx} + y = x\cos(x)$ , for $x > 0$ .

Given that $y = \frac{2}{\pi}$ when $x = \pi$ , find the solution in the form $y = f(x)$ .

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What does Linear first order differential equations cover in IB Maths AA?

The Maclaurin series is a Taylor series expansion of a function f(x) centered at a=0. It expresses f(x) as an infinite sum of terms involving its derivatives evaluated at x=0. The general formula is f(x) = Σ_n=0^∞ fracf^(n)(0)n! x^n.

Is Linear first order differential equations SL or HL?

Linear first order differential equations is HL only. SL students are not examined on it.

How do I revise Linear first order differential equations for IB Maths AA?

Start from the core idea: the Maclaurin series is a Taylor series expansion of a function f(x) centered at a=0. In the exam: four methods under one code, and the question usually names the method, so generation should name it too. The logistic equation is the IB's own example and it pulls in partial fractions, which makes it a good multi-part question. Then practise exam-style questions, easiest first, writing out every step of your working before you check it.

How does FourtyFive help me practise Linear first order differential equations?

FourtyFive has 7 Linear first order differential equations questions. Every answer you write is marked mark by mark, IB-style, and you see where each mark was won or lost. Every part has a hint, the AI tutor helps you through the step you are stuck on, and your Study Profile picks what to practise next.

Is FourtyFive free for Linear first order differential equations practice?

Yes. A free account gives you 50 marked answers a month, and you do not need a card to sign up.

Can I handwrite Linear first order differential equations answers on an iPad?

Yes. In the FourtyFive iPad app you write your working by hand with Apple Pencil, the way you would on paper, and it is marked the same way.

Linear first order differential equations: notes and practice questions

How it is examined

Linking questions

Practice questions

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Every Linear first order differential equations question, marked for you

Where marks are lost

Practise this topic as a session

Questions,
answered.

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How it is examined

Linking questions

Practice questions

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Every Linear first order differential equations question, marked for you

Where marks are lost

Questions,answered.

Start with the IB questionbank built for you.

Questions,
answered.

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