Numerical Solutions to Differential Equations (DP IB Maths: AA HL)

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Roger

Last updated

31 October 2023

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First Order Differential Equations

What is a differential equation?

A differential equation is simply an equation that contains derivatives

For example $\frac{d y}{d x} = 12 x y^{2}$ is a differential equation
And so is $\frac{d^{2} x}{d t^{2}} - 5 \frac{d x}{d t} + 7 x = 5 \sin t$

What is a first order differential equation?

A first order differential equation is a differential equation that contains first derivatives but no second (or higher) derivatives

For example $\frac{d y}{d x} = 12 x y^{2}$ is a first order differential equation
But $\frac{d^{2} x}{d t^{2}} - 5 \frac{d x}{d t} + 7 x = 5 \sin t$ is not a first order differential equation, because it contains the second derivative $\frac{d^{2} x}{d t^{2}}$

Wait – haven’t I seen first order differential equations before?

Yes you have!

For example $\frac{d y}{d x} = 3 x^{2}$ is also a first order differential equation, because it contains a first derivative and no second (or higher) derivatives
But for that equation you can just integrate to find the solution y = x³ + c (where c is a constant of integration)

In this section of the course you learn how to solve differential equations that can’t just be solved right away by integrating

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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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Numerical Solutions to Differential Equations (DP IB Maths: AA HL)

Revision Note

What is a differential equation?

What is a first order differential equation?

Wait – haven’t I seen first order differential equations before?

What is Euler’s method?

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

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

Sign up now. It’s free!

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

1. Number & Algebra

1.1 Number & Algebra Toolkit

1.1.1 Standard Form

1.1.2 Laws of Indices

1.1.3 Partial Fractions

1.2 Exponentials & Logs

1.2.1 Introduction to Logarithms

1.2.2 Laws of Logarithms

1.2.3 Solving Exponential Equations

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

1.3.5 Compound Interest & Depreciation

1.4 Simple Proof & Reasoning

1.4.1 Proof

1.5 Further Proof & Reasoning

1.5.1 Proof by Induction

1.5.2 Proof by Contradiction

1.6 Binomial Theorem

1.6.1 Binomial Theorem

1.6.2 Extension of The Binomial Theorem

1.7 Permutations & Combinations

1.7.1 Counting Principles

1.7.2 Permutations & Combinations

1.8 Complex Numbers

1.8.1 Intro to Complex Numbers

1.8.2 Modulus & Argument

1.8.3 Introduction to Argand Diagrams

1.9 Further Complex Numbers

1.9.1 Geometry of Complex Numbers

1.9.2 Forms of Complex Numbers

1.9.3 Complex Roots of Polynomials

1.9.4 De Moivre's Theorem

1.9.5 Roots of Complex Numbers

1.10 Systems of Linear Equations

1.10.1 Systems of Linear Equations

1.10.2 Algebraic Solutions

2. Functions

2.1 Linear Functions & Graphs

2.1.1 Equations of a Straight Line

2.2 Quadratic Functions & Graphs

2.2.1 Quadratic Functions

2.2.2 Factorising & Completing the Square

2.2.3 Solving Quadratics

2.2.4 Quadratic Inequalities

2.2.5 Discriminants

2.3 Functions Toolkit

2.3.1 Language of Functions

2.3.2 Composite & Inverse Functions

2.3.3 Symmetry of Functions

2.3.4 Graphing Functions

2.4 Other Functions & Graphs

2.4.1 Exponential & Logarithmic Functions

2.4.2 Solving Equations

2.4.3 Modelling with Functions

2.5 Reciprocal & Rational Functions

2.5.1 Reciprocal & Rational Functions

2.6 Transformations of Graphs

2.6.1 Translations of Graphs

2.6.2 Reflections of Graphs

2.6.3 Stretches Graphs

2.6.4 Composite Transformations of Graphs

2.7 Polynomial Functions

2.7.1 Factor & Remainder Theorem

2.7.2 Polynomial Division

2.7.3 Polynomial Functions

2.7.4 Roots of Polynomials

2.8 Inequalities