Lesson 4 - Euler's Method

Euler's method, or the tangent line method, is used to construct approximate solutions to a first order differential equation of the form $\frac{d y}{d x} = f (x, y)$ , $y (x_{0}) = y_{0}$ .

We assume that the IVP has a unique solution in some interval centered at $x_{0}$ . $h > 0$ is a positive real number called the step size.

Euler Method.png

We pick equally spaces points on the x axis: $x_{0}, x_{0} + h, x_{0} + 2 h, \dots$ . In general, $x_{n} = x_{0} + n h, n = 0, 1, 2, 3, \dots$ .
We start by finding the slope at the first point $(x_{0}, y_{0})$ which is $f (x_{0}, y_{0})$ . We trace the line with this slope until the next point $(x_{1}, y_{1})$ when reset the slope to the slope at $(x_{1}, y_{1})$ : $f (x_{1}, y_{1})$ and follow that line until the next point. We repeat this process.
Equation of line at $(x_{0}, y_{0})$ is

y - b = m (x - a) \to y - y_{0} = m (x - x_{0}) \to y - y_{0} = f (x_{0}, y_{0}) (x - x_{0})

Using the tangent line approximation find a formula for $y_{1}$

y_{1} = y_{0} + f (x_{0}, y_{0}) (x_{1} - x_{0}) \to y_{1} = y_{0} + h f (x_{0}, y 0)

Using the above method find a formula for $y_{2}$

y_{2} = y_{1} + h (f (x_{1}, y_{1}))

Can we deduce a general formula for $y_{n}$ ?

y_{n} = y_{n - 1} + h (f (x_{n - 1}, y_{n - 1})), n = 0, 1, 2, 3, \dots

x_{n + 1} = x_{n} + h

Example: Use Euler's method to find approximate value of $y (0.4)$ where $y$ is the solution of the IVP $y^{'} = y, y (0) = 1$ .

s l o p e = y

y_{n} = y_{n - 1} + h f (x_{n - 1} y_{n - 1})

n = 0

h = 0.1

y_{1} = 1 + 0.1 (1) = 1.1

y_{2} = 1.1 + 0.1 (1.1)

y_{2} = 1.21

y_{3} = 1.21 + 0.1 (1.21)

y_{3} = 1.331

y_{4} = 1.331 + 0.1 (1.331)

y_{4} = 1.4641

Try For Yourself: Use Euler's method to approximate the solution to the IVP.

$y^{'} = x - y^{2}, y (1) = 0$ at the point $x = 1.1, 1.2, 1.3, 1.4, 1.5$ .

Next Lesson: Lesson 5 - Separable Equations

Table of Contents: Table of Contents