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Eulers Method

Leonhard Euler made a huge number of

contributions to mathematics, almost half after

he was totally blind.

(When this portrait was made he had already lost

most of the sight in his right eye.)

Leonhard Euler 1707 - 1783

Greg Kelly, Hanford High School, Richland,

Washington

It was Euler who originated the following

notations

Leonhard Euler 1707 - 1783

Recall a differential equation problem

Find the particular solution

that satisfies that initial condition (1, 0)

There are many differential equations that can

not be solved. But we can still find an

approximate solution at particular points.

Euler came up with a method that we can use to do

this.

Essentially what we will do is 1) Start with an

initial value, 2) Use the slope to find a

tangent line at that point, 3) Then use the

tangent line to approximate another value on the

curve. We can repeat this process as many times

as necessary.

Ex. 1 Use Eulers Method to find the first three

approximations of the differential equation

passing through (0, 1).

Use a step of h 0.1

Tangent line at (0, 1)

Since step 0.1,

Use tangent line to get

Tangent line at (0.1, 0.9)

Since step 0.1,

Use tangent line to get

Tangent line at (0.2, 0.82)

Since step 0.1,

Use tangent line to get

Third approximation (0.3, 0.758)

Ex. 2

(No Transcript)

Exact Solution

(No Transcript)

Ex. 3

The table gives selected values of the derivative

of a function g at certain x-values. If

and Eulers method with a step size of

1.5 is used to approximate then what is

the resulting approximation?

x g(x)

-1.0 2

-0.5 4

0.0 3

0.5 1

1.0 0

1.5 -3

2.0 -6

Initial value (-1, -2)

Slope there 2

At (0.5, 1) Slope there 1

It took us 2 iterations to get there!

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