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

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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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