Euler’s Method
Lesson 2.1
Objective
• Apply Euler’s method to approximate differential equations
Definitions
Euler’s Method – A way to approximate differential equations
using successive linear approximations.
Euler’s Method
The method is based on the idea of linear approximation.
𝑓 𝑥 ≈ 𝑓′ 𝑐 𝑥 − 𝑐 + 𝑓 𝑐
Or
𝑦 = 𝑚 𝑥 − 𝑥1 + 𝑦1
However, with Euler’s method we don’t stay on the same line.
We reformulate the tangent line as we take steps 𝑑𝑥 .
Euler’s Method
To help apply Euler’s method we will use the following table:
1) Applying Euler’s Method
𝑑𝑦
Consider the differential equation = 𝑥 + 𝑥𝑦. Let 𝑦 = 𝑓(𝑥) be the
𝑑𝑥
function that satisfies the given differential equation with the initial
condition 𝑓(1) = 2. Use Euler’s method, starting at 𝑥 = 1 with a step size
of 0.5, to approximate 𝑓(2).
𝒙
𝒚
𝒎=
𝒅𝒚
𝒅𝒙
𝑑𝑦 = 𝑚 ∙ 𝑑𝑥
𝑦𝑛𝑒𝑤 = 𝑦 + 𝑑𝑦
2) Applying Euler’s Method
𝑑𝑦
Consider the differential equation = 𝑦 + 𝑥 2 . Let 𝑦 = 𝑓(𝑥) be the
𝑑𝑥
function that satisfies the given differential equation with the initial
condition 𝑓(0) = 1. Use Euler’s method, starting at 𝑥 = 0 and using 2
steps, to approximate 𝑓(2).
𝒙
𝒚
𝒎=
𝒅𝒚
𝒅𝒙
𝑑𝑦 = 𝑚 ∙ 𝑑𝑥
𝑦𝑛𝑒𝑤 = 𝑦 + 𝑑𝑦
3) Applying Euler’s Method
𝑑𝑦
Consider the differential equation = 𝑥 2 + 𝑥 − 1. Let 𝑦 = 𝑓(𝑥) be the
𝑑𝑥
function that satisfies the given differential equation with the initial
condition 𝑓(0) = 1. Use Euler’s method, starting at 𝑥 = 0 with a step size
of 0.5, to approximate 𝑓(1).
𝒅𝒚
𝒙
𝒚
𝑑𝑦 = 𝑚 ∙ 𝑑𝑥
𝑦𝑛𝑒𝑤 = 𝑦 + 𝑑𝑦
𝒎=
𝒅𝒙
4) Applying Euler’s Method
The rate at which the flu spreads through a community is modeled by the
𝑑𝑃
logistic differential equation = 0.001𝑃 3000 − 𝑃 , where 𝑡 is measured
𝑑𝑡
in days and 𝑡 ≥ 0. Using Euler’s method with 3 steps, approximate the
population infected after 3 days if the initial population infected is 50.
𝒙
𝒚
𝒎=
𝒅𝒚
𝒅𝒙
𝑑𝑦 = 𝑚 ∙ 𝑑𝑥
𝑦𝑛𝑒𝑤 = 𝑦 + 𝑑𝑦
5) Applying Euler’s Method
5) Applying Euler’s Method