Topic 5.17 · HL only

Euler method for Second order differential equations: notes and practice questions

Summary

Euler's Method: Numerical approximation for differential equations using step-by-step calculations.
Step Size ( $h$ ): Constant increment for the independent variable ( $t$ ).
Second Order Differential Equations: Must be transformed into a system of two coupled first order differential equations.
Substitution Rule:
Let $y = \frac{dx}{dt}$
Then $\frac{dy}{dt} = \frac{d^2x}{dt^2}$
Euler's Method for Coupled Systems (for $\frac{dx}{dt} = f_1(x, y, t)$ and $\frac{dy}{dt} = f_2(x, y, t)$ ):
$x_{n+1} = x_n + h \times f_1(x_n, y_n, t_n)$
$y_{n+1} = y_n + h \times f_2(x_n, y_n, t_n)$
$t_{n+1} = t_n + h$
Procedure:

1. Substitute $y = \frac{dx}{dt}$ and $\frac{dy}{dt} = \frac{d^2x}{dt^2}$ into the second order DE.
2. Form the coupled system: $\frac{dx}{dt} = y$ and $\frac{dy}{dt} = \text{rearranged equation}$ .
3. Identify initial conditions ( $x_0, y_0, t_0$ ), where $y_0$ is the initial value of $\frac{dx}{dt}$ .
4. Set up recursion equations using Euler's formulas with $f_1, f_2, h$ .
5. Iterate using a GDC's recursion feature.

HL Only: Solving differential equations (first order, coupled, second order) and applying Euler's Method are HL topics.
Improving Accuracy: Decrease the step size ( $h$ ).
Contextual Interpretation: If $x$ is displacement, then $y = \frac{dx}{dt}$ represents velocity.
GDC Settings: Use radians mode for equations involving trigonometric functions.

How it is examined

The whole subtopic hinges on the rewrite: once the second order equation is expressed as a coupled first order system, it is worked exactly like AHL 5.16 (Euler's method) or AHL 5.17 (eigenvalues and phase portrait), so the method mark for "correctly writes the coupled system" tends to come before any marks for the numerical or exact solution itself. This is the last subtopic in the course, and it is deliberately a synthesis of three subtopics rather than new content of its own.

Key ideas

Solve $\dfrac{d^2x}{dt^2} = f\left(x, \dfrac{dx}{dt}, t\right)$ by Euler's method, writing it as the coupled first order system $\dfrac{dx}{dt} = y$ , $\dfrac{dy}{dt} = f(x, y, t)$ .
Solve $\dfrac{d^2x}{dt^2} + a\dfrac{dx}{dt} + bx = 0$ by finding exact solutions, using the phase portrait method of AHL 5.17.

Not assessed

In examinations the second order equation will be given, so a student is not expected to set one up from a physical context, only to solve it once stated.

Linking questions

TOK: how have notable individuals such as Euler shaped the development of mathematics as an area of knowledge?
Use of technology: spreadsheets to generate values.

Practice questions

4 questions · 3 medium · 1 hard

Showing 4 of 4

Question 1

MediumPaper 1 · calculator6 marks

If a vehicle is moving along a straight road, where its position $y$ in $cm$ at time $t$ in seconds for $t \geq 0$ is given by the following differential equation:

$3\frac{d^{2}y}{dt^{2}} + 8\frac{dy}{dt} + 5y = 12sin(2t)$

At $t = 0$ take $y = 1$ cm and $\frac{dy}{dt} = 0.2$ . Use Euler's method with a step size of 0.1 and estimate $\frac{dy}{dt}$ at $t = 0.4$ .

Question 2

HardPaper 2 · calculator16 marks

(a)

A chemical engineer is studying the concentration of an intermediate product, $C$ , in a reaction vessel. The change in concentration over time $t$ (in minutes) is modelled by the second order differential equation:

$\frac{d^2C}{dt^2} + 4\frac{dC}{dt} + 3C = 0$

where $t \ge 0$ . It is known that when $t = 0$ , the initial concentration is $C = 0$ and the rate of change of concentration is $\frac{dC}{dt} = 2$ .

Show that the system of coupled first order equations:

$\frac{dC}{dt} = y$

$\frac{dy}{dt} = -3C - 4y$

can be written as the given second order differential equation.

[2]

(b)

Find the eigenvalues of the system of coupled first order equations given in part (a).

[3]

(c)

Hence find the exact solution of the second order differential equation, given the initial conditions $C(0) = 0$ and $\frac{dC}{dt}(0) = 2$ .

[5]

(d)

Sketch the graph of $C$ against $t$ for $t \ge 0$ , labelling the maximum point of the graph with its coordinates.

[2]

(e)

If the concentration of the intermediate product $C$ exceeds $0.2$ arbitrary units, the reaction needs to be monitored closely. Use the model to calculate the total amount of time (in minutes) during which the reaction needs to be monitored closely.

[3]

(f)

The chemical engineer decides to monitor the reaction for 15% longer than the time found from the model in part (e).

Write down one reason, with reference to the context, to support this decision.

[1]

Question 3

MediumPaper 1 · calculator6 marks

If a boat is moving in a straight line, where its position $y$ in $m$ at time $t$ in seconds for $t \geq 0$ is given by the following differential equation:

$3\frac{d^{2}y}{dt^{2}} + 9\frac{dy}{dt} + 6y = sin(4t)$

At $t = 0$ take $y = 2$ m and $\frac{dy}{dt} = 1$ . Use Euler's method with a step size of 0.05 and estimate $\frac{dy}{dt}$ at $t = 0.2$ .

Question 4

MediumPaper 2 · calculator20 marks

(a)

If a particle moves in a magnetic field and its motion is described by its displacement $x$ (in meters) from equilibrium position. If the magnetic force affects the motion $x$ of the particle and leads to the following second order differential equation:

$\ddot{x} + 6\dot{x} + 8x = 0$

Where $t$ is time in seconds.

$a$ Take $\dot{x} = y$ and find an expression for $\dot{y}$ in terms of $x$ and $y$ .

[2]

(b)

$b$ Find $A$ such that $\begin{pmatrix} \dot{x} \\ \dot{y} \end{pmatrix} = A\begin{pmatrix} x \\ y \end{pmatrix}$ , where $A$ is a $2 \times 2$ matrix.

[1]

(c)

$c$ Find the eigenvalues and eigen vectors of matrix $A$ .

[6]

(d)

$d$ Determine whether the equilibrium point (0,0) is stable or unstable.

[2]

(e)

$e$ Find $\frac{dy}{dx}$ at the point (1,2) and (3,2).

[3]

(f)

If the equation is amended to the following:

$\ddot{x} + 6\dot{x} + 8x = 5t + 1$

$f$ At $t = 0$ take $x = 0$ cm and $\dot{x} = 0$ . Use Euler's method with a step length of 0.25 to find the value of $x$ at $t = 1$ .

[6]

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What does Euler method for Second order differential equations cover in IB Maths AI?

Euler's Method: Numerical approximation for differential equations using step-by-step calculations. Step Size (h): Constant increment for the independent variable (t). Second Order Differential Equations: Must be transformed into a system of two coupled first order differential equations.

Is Euler method for Second order differential equations SL or HL?

Euler method for Second order differential equations is HL only. SL students are not examined on it.

How do I revise Euler method for Second order differential equations for IB Maths AI?

Start from the core idea: euler's Method: Numerical approximation for differential equations using step-by-step calculations. In the exam: the whole subtopic hinges on the rewrite: once the second order equation is expressed as a coupled first order system, it is worked exactly like AHL 5.16 (Euler's method) or AHL 5.17 (eigenvalues and phase portrait), so the method mark for "correctly writes the coupled system" tends to come before any marks for the numerical or exact solution itself. This is the last subtopic in the course, and it is deliberately a synthesis of three subtopics rather than new content of its own. Then practise exam-style questions, easiest first, writing out every step of your working before you check it.

How does FourtyFive help me practise Euler method for Second order differential equations?

FourtyFive has 4 Euler method for Second 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.

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Euler method for Second order differential equations: notes and practice questions

How it is examined

Linking questions

Practice questions

Question 1

Question 2

Question 3

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Every Euler method for Second order differential equations question, marked for you

Where marks are lost

Practise this topic as a session

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How it is examined

Linking questions

Practice questions

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Question 3

Question 4

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Where marks are lost

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