Approximate Solutions to Differential Equations (DP IB Maths: AI HL)

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Author

Roger

Last updated

1 November 2023

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Euler’s Method: First Order

What is Euler’s method?

Euler’s method is a numerical method for finding approximate solutions to differential equations
It treats the derivatives in the equation as being constant over short ‘steps’
The accuracy of the Euler’s Method approximation can be improved by making the step sizes smaller

How do I use Euler’s method with a first order differential equation?

STEP 1: Make sure your differential equation is in $\frac{d y}{d x} = f (x, y)$ form
STEP 2: Write down the recursion equations using the formulae $y_{n + 1} = y_{n} + h \times f (x_{n}, y_{n})$ and $x_{n + 1} = x_{n} + h$ from the exam formula booklet

h in those equations is the step size
the exam question will usually tell you the correct value of h to use

STEP 3: Use the recursion feature on your GDC to calculate the Euler’s method approximation over the correct number of steps

the values for $x_{0}$ and $y_{0}$ will come from the boundary conditions given in the question

Examiner Tip

Be careful with letters – in the equations in the exam, and in your GDC’s recursion calculator, the variables may not be x and y
If an exam question asks you how to improve an Euler’s method approximation, the answer will almost always have to do with decreasing the step size!

Worked example

Consider the differential equation $\frac{d y}{d x} + y = x + 1$ with the boundary condition $y (0) = 0.5$ .

Apply Euler’s method with a step size of $h = 0.2$ to approximate the solution to the differential equation at $x = 1$ .

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Explain how the accuracy of the approximation in part (a) could be improved.

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Euler's Method: Coupled Systems

How do I use Euler’s method with coupled first order differential equations?

STEP 1: Make sure your coupled differential equations are in $\frac{d x}{d t} = f_{1} (x, y, t)$ and $\frac{d y}{d t} = f_{2} (x, y, t)$ form
STEP 2: Write down the recursion equations using the formulae $x_{n + 1} = x_{n} + h \times f_{1} (x_{n}, y_{n}, t_{n})$ , $y_{n + 1} = y_{n} + h \times f_{1} (x_{n}, y_{n}, t_{n})$ and $t_{n + 1} = t_{n} + h$ from the exam formula booklet

h in those equations is the step size
the exam question will usually tell you the correct value of h to use

STEP 3: Use the recursion feature on your GDC to calculate the Euler’s method approximation over the correct number of steps

the values for $x_{0}$ , $y_{0}$ and $t_{0}$ will come from the boundary conditions given in the question
frequently you will be given an initial condition

look out for terms like ‘initially’ or ‘at the start’
in this case $t_{0}$

Examiner Tip

Be careful with letters – in the equations in the exam, and in your GDC’s recursion calculator, the variables may not be x, y and t.
If an exam question asks you how to improve an Euler’s method approximation, the answer will almost always have to do with decreasing the step size!

Worked example

Consider the following system of differential equations:

$\frac{d x}{d t} = 2 x - 3 y + 1$

$\frac{d y}{d t} = x + y + \frac{1}{t + 1}$

Initially x = 10 and y = 2.

Use Euler’s method with a step size of 0.1 to find approximations for the values of x and y when t = 0.5.

5-6-4-ib-ai-hl-eulers-method-coupled-we-solution

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Previous:5.6.3 Slope FieldsNext:5.7.1 Coupled Differential Equations

Approximate Solutions to Differential Equations (DP IB Maths: AI HL)

Revision Note

What is Euler’s method?

How do I use Euler’s method with a first order differential equation?

How do I use Euler’s method with coupled first order differential equations?

You've read 0 of your 5 free revision notes this week

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Join the 100,000+ Students that ❤️ Save My Exams

1. Number & Algebra

1.1 Number Toolkit

1.1.1 Standard Form

1.1.2 Approximation & Estimation

1.1.3 GDC: Solving Equations

1.2 Exponentials & Logs

1.2.1 Exponents

1.2.2 Logarithms

1.3 Sequences & Series

1.3.1 Language of Sequences & Series

1.3.2 Arithmetic Sequences & Series

1.3.3 Geometric Sequences & Series

1.3.4 Applications of Sequences & Series

1.4 Financial Applications

1.4.1 Compound Interest & Depreciation

1.4.2 Amortisation & Annuities

1.5 Complex Numbers

1.5.1 Intro to Complex Numbers

1.5.2 Modulus & Argument

1.5.3 Introduction to Argand Diagrams

1.6 Further Complex Numbers

1.6.1 Geometry of Complex Numbers

1.6.2 Forms of Complex Numbers

1.6.3 Applications of Complex Numbers

1.7 Matrices

1.7.1 Introduction to Matrices

1.7.2 Operations with Matrices

1.7.3 Determinants & Inverses

1.7.4 Solving Systems of Linear Equations with Matrices

1.8 Eigenvalues & Eigenvectors

1.8.1 Eigenvalues & Eigenvectors

1.8.2 Applications of Matrices

2. Functions

2.1 Linear Functions & Graphs

2.1.1 Equations of a Straight Line

2.2 Further Functions & Graphs

2.2.1 Functions

2.2.2 Graphing Functions

2.2.3 Properties of Graphs

2.3 Modelling with Functions

2.3.1 Linear Models

2.3.2 Quadratic & Cubic Models

2.3.3 Exponential Models

2.3.4 Direct & Inverse Variation

2.3.5 Sinusoidal Models

2.3.6 Strategy for Modelling Functions

2.4 Functions Toolkit

2.4.1 Composite & Inverse Functions

2.5 Transformations of Graphs

2.5.1 Translations of Graphs

2.5.2 Reflections of Graphs

2.5.3 Stretches of Graphs

2.5.4 Composite Transformations of Graphs

2.6 Further Modelling with Functions

2.6.1 Properties of Further Graphs

2.6.2 Natural Logarithmic Models

2.6.3 Logistic Models

2.6.4 Piecewise Models

3. Geometry & Trigonometry

3.1 Geometry Toolkit

3.1.1 Coordinate Geometry

3.1.2 Radian Measure

3.1.3 Arcs & Sectors

3.2 Geometry of 3D Shapes

3.2.1 3D Coordinate Geometry

3.2.2 Volume & Surface Area

3.3 Trigonometry

3.3.1 Pythagoras & Right-Angled Trigonometry

3.3.2 Non Right-Angled Trigonometry

3.3.3 Applications of Trigonometry & Pythagoras

3.4 Further Trigonometry

3.4.1 The Unit Circle