Slope Fields and Euler’s Method

• When taking an antiderivative that is not dealing with a definite integral, be sure to add the constant at the end.

Given: 2y x find y

2y x C We don’t know what the constant is, so we put “C” in the answer to remind us that there might have been a constant.

If we have some more information we can find C.

Given: y’ = 2x and f(1) = 4, find the equation for y.

2y x C 24 1 C

3 C2 3y x

This is called an initial value problem. We need the initial values to find the constant.

An equation containing a derivative is called a differential equation. It becomes an initial value problem when you are given the initial condition and asked to find the original equation.

Initial value problems and differential equations can be illustrated with a slope field.

Slope fields are mostly used as a learning tool and are mostly done on a computer or graphing calculator, but recent AP tests have asked students to draw a simple one by hand.

Draw a segment with slope of 2.

Draw a segment with slope of 0.

Draw a segment with slope of 4.

2y x

x y y0 0 00 1 00 00 0

23

1 0 21 1 2

2 0 4

-1 0 -2

-2 0 -4

2y x

If you know an initial condition, such as (1,-2), you can sketch the curve.

By following the slope field, you get a rough picture of what the curve looks like.

In this case, it is a parabola.

Integrals such as are called indefinite integrals

because we can not find a definite value for the answer.

2x dx

2x dx31

3x C

When finding indefinite integrals, we always include the “plus C”.

Euler’s Method

Leonhard Euler 1707 - 1783

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

It was Euler who originated the following notations:

e (base of natural log)

f x (function notation)

(pi)

i 1

(summation)

y (finite change)

There are many differential equations that can not be solved.We can still find an approximate solution.

We will practice with an easy one that can be solved.

2dy

xdx

Initial value: 0 1y

Steps for using Euler’s Method:• Begin a the point specified by the initial condition.

• Use the d.e. to find the slope dy/dx at that point.

• Increase x by a small amount Δx. Increase y by a small amount Δy where Δy = (dydx)Δx. This defines a new point (x + Δx, y + Δy) that lies along the linearization.

• Use this new point, return to step 2. Repeating the process constructs the graph to the right of the initial point.

2dy

xdx

0 1y

n nx nydy

dx dy 1ny

0.5dx

0 0 1 0 0 1

1 .5 1 1 .5 1.5

2 1 1.5 2 1 2.5

dydx dy

dx 1n ny dy y

3 1.5 2.5 3 1.5 4.0

4 2.0 4.0

dydx dy

dx 1n ny dy y

2dy

xdx

n nx nydy

dx dy 1ny

0.5dx

0 0 1 0 0 1

1 .5 1 1 .5 1.5

2 1 1.5 2 1 2.5

0 1y

2dy

xdx

0,1 0.5dx

2 dy x dx

2y x C

1 0 C

2 1y x

Exact Solution:

It is more accurate if a smaller value is used for dx.

It gets less accurate as you move away from the initial value.

Examples 9 and 10 also show Euler’s Method

I also have a program that will do an Euler Table and graph.