EULER’S METHOD

Section 6.1A

Calculus BC AP/Dual, Revised ©2018

viet.dang@humbleisd.net

8/7/2018 12:51 AM §6.1A: Euler's Method 1

ABOUT EULER

A. Leonhard Euler1. Pronounced as “Oiler”

2. Made a huge number of contributions to mathematics, almost half after he was totally blind.

3. Lived from 1707-83

4. Created such notations such asA. Function Notation, 𝒇(𝒙)

B. Base of Natural Log, e

C. Pi, 𝝅

D. Imaginary Number, i

E. Summation Notation, 𝚺

8/7/2018 12:51 AM §6.1A: Euler's Method 2

CONSTRUCTING EULER’S METHOD

A. Euler’s Method for Graphing a Solution to an Initial Value Problem or to approximate solutions of differential equations

B. Apply the equation, 𝒀𝒏 = 𝒀𝟏 +𝒎 𝒙 − 𝒙𝟏 or written as:

𝒀 = 𝒀𝟏 +𝒅𝒚

𝒅𝒙Δ𝒙

1. 𝒀𝒏 = New 𝒀-Coordinate

2. 𝒀𝟏 = Previous/Initial 𝒀-Coordinate

3. 𝑿𝟏 = Previous/Initial 𝑿-Coordinate

4. 𝒎 = Step Change

8/7/2018 12:51 AM §6.1A: Euler's Method 3

PROCESS

8/7/2018 12:51 AM §6.1A: Euler's Method 4

,x y

,x x y y

x

y

dyx y

dx •

slope

dy y

dx x

1 1y y m x x

THE EQUATION

8/7/2018 12:51 AM §6.1A: Euler's Method 5

dy

dxx 1

dyy y x

dx

STEPS

A. Find the slope 𝒅𝒚

𝒅𝒙at the initial point.

B. Plug in the point given into the equation

C. Establish the new equation

D. Use the old 𝒙 coordinate to determine the NEW slope and repeat the process until the desired point is established.

E. To construct the graph to the left from the initial point, repeat the process using negative values for 𝚫𝒙.

8/7/2018 12:51 AM §6.1A: Euler's Method 6

EXAMPLE 1

Use Euler’s Method to approximate the particular solution of the

differential equation 𝒚′ =𝒙

𝟐passing through the point 𝒇 𝟎 = 𝟏,

using a step size of Δ𝒙 = 𝟏, and approximate 𝒚 𝟑 using 𝟑 steps.

8/7/2018 12:51 AM §6.1A: Euler's Method 7

𝒙 𝒚 𝒅𝒚

𝒅𝒙∆𝒙

𝟎

final y initial ytotal steps

x

2

2

2

03

1

0 3

3

y

y

y

𝟎 𝟏𝒅𝒚

𝒅𝒙=𝟎

𝟐→ 𝟎 𝟏 = 𝟎

𝟏0

02 2

dy x

dx

EXAMPLE 1

Use Euler’s Method to approximate the particular solution of the

differential equation 𝒚′ =𝒙

𝟐passing through the point 𝒇 𝟎 = 𝟏,

using a step size of Δ𝒙 = 𝟏, and approximate 𝒚 𝟑 using 𝟑 steps.

8/7/2018 12:51 AM §6.1A: Euler's Method 8

𝒙 𝒚 𝒅𝒚

𝒅𝒙∆𝒙

𝟎 𝟏𝟎

𝟐→ 𝟎 𝟑 = 𝟎

𝟏

'2

xy 0 1f 1x

1

dyy y x

dx

1 0 1y

1 0y 1 1y 1,1

2

dy x

dx

00

2

𝟎 𝟏𝒅𝒚

𝒅𝒙=𝟎

𝟐→ 𝟎 𝟏 = 𝟎

𝟏 𝟏

𝟐

EXAMPLE 1

Use Euler’s Method to approximate the particular solution of the

differential equation 𝒚′ =𝒙

𝟐passing through the point 𝒇 𝟎 = 𝟏,

using a step size of Δ𝒙 = 𝟏, and approximate 𝒚 𝟑 using 𝟑 steps.

8/7/2018 12:51 AM §6.1A: Euler's Method 9

0

1

2

3

4

5

1 2 3

𝒙 𝒚 𝒅𝒚

𝒅𝒙∆𝒙

𝟎 𝟏 𝟎

𝟏 𝟏

𝟐

EXAMPLE 1

Use Euler’s Method to approximate the particular solution of the

differential equation 𝒚′ =𝒙

𝟐passing through the point 𝒇 𝟎 = 𝟏,

using a step size of Δ𝒙 = 𝟏, and approximate 𝒚 𝟑 using 𝟑 steps.

8/7/2018 12:51 AM §6.1A: Euler's Method 10

1 1

2 2 2

dy x

dx

𝒙 𝒚 𝒅𝒚

𝒅𝒙∆𝒙

𝟎 𝟏 𝟎

𝟏 𝟏

𝟎 𝟏 𝟎

𝟏 𝟏𝒅𝒚

𝒅𝒙=𝟏

𝟐→𝟏

𝟐𝟏 =

𝟏

𝟐

𝟐

Step Size 1

EXAMPLE 1

Use Euler’s Method to approximate the particular solution of the

differential equation 𝒚′ =𝒙

𝟐passing through the point 𝒇 𝟎 = 𝟏,

using a step size of Δ𝒙 = 𝟏, and approximate 𝒚 𝟑 using 𝟑 steps.

8/7/2018 12:51 AM §6.1A: Euler's Method 11

𝒙 𝒚 𝒅𝒚

𝒅𝒙∆𝒙

𝟎 𝟏 𝟎

𝟏 𝟏𝒅𝒚

𝒅𝒙=𝟏

𝟐→𝟏

𝟐𝟏 =

𝟏

𝟐

𝟐

'2

xy 0 1f 1x

2

dy x

dx

00

2

1

dyy y x

dx

1

1 12

y

11

2y

3

2y

32,

2

𝟎 𝟏 𝟎

𝟏 𝟏𝟏

𝟐

𝟐𝟑

𝟐

𝟑

EXAMPLE 1

Use Euler’s Method to approximate the particular solution of the

differential equation 𝒚′ =𝒙

𝟐passing through the point 𝒇 𝟎 = 𝟏,

using a step size of Δ𝒙 = 𝟏, and approximate 𝒚 𝟑 using 𝟑 steps.

8/7/2018 12:51 AM §6.1A: Euler's Method 12

0

1

2

3

4

5

1 2 3

𝒙 𝒚 𝒅𝒚

𝒅𝒙∆𝒙

𝟎 𝟏 𝟎

𝟏 𝟏𝟏

𝟐

𝟐𝟑

𝟐

𝟑

EXAMPLE 1

Use Euler’s Method to approximate the particular solution of the

differential equation 𝒚′ =𝒙

𝟐passing through the point 𝒇 𝟎 = 𝟏,

using a step size of Δ𝒙 = 𝟏, and approximate 𝒚 𝟑 using 𝟑 steps.

8/7/2018 12:51 AM §6.1A: Euler's Method 13

𝒙 𝒚 𝒅𝒚

𝒅𝒙∆𝒙

𝟎 𝟏 𝟎

𝟏 𝟏𝟏

𝟐

𝟐𝟑

𝟐

𝟑

𝟎 𝟏 𝟎

𝟏 𝟏𝟏

𝟐

𝟐𝟑

𝟐

𝒅𝒚

𝒅𝒙=𝟐

𝟐→ 𝟏 𝟏 = 𝟏

𝟑

3

1 12

y

31

2y

53,

2

5

32

y

'2

xy 0 1f 1x

2

dy x

dx

21

2

1

dyy y x

dx 𝟎 𝟏 𝟎

𝟏 𝟏𝟏

𝟐

𝟐𝟑

𝟐 𝟏

𝟑𝟓

𝟐

x

y

0

1

2

3

4

5

1 2 3

EXAMPLE 1

Use Euler’s Method to approximate the particular solution of the

differential equation 𝒚′ =𝒙

𝟐passing through the point 𝒇 𝟎 = 𝟏,

using a step size of Δ𝒙 = 𝟏, and approximate 𝒚 𝟑 using 𝟑 steps.

8/7/2018 12:51 AM §6.1A: Euler's Method 14

5

32

y 𝒙 𝒚 𝒅𝒚

𝒅𝒙∆𝒙

𝟎 𝟏 𝟑𝟎

𝟐= 𝟎

𝟏 𝟏 𝟑𝟏

𝟐

𝟐𝟑

𝟐 𝟑 𝟏

𝟑

𝟎 𝟏 𝟎

𝟏 𝟏𝟏

𝟐

𝟐𝟑

𝟐 𝟏

𝟑𝟓

𝟐

YOUR TURN

Use Euler’s Method to approximate the particular solution of the differential equation 𝒚′ = 𝟐𝒙 passing through the point 𝒇 𝟎 =𝟏 and using a step size of Δ𝒙 = 𝟎. 𝟓, and approximate 𝒚 𝟐 using 4 steps.

8/7/2018 12:51 AM §6.1A: Euler's Method 15

2 4y

0

1

2

3

4

5

1 2 3

YOUR TURN

8/7/2018 12:51 AM §6.1A: Euler's Method 16

2 4y

Use Euler’s Method to approximate the particular solution of the differential equation 𝒚′ = 𝟐𝒙 passing through the point 𝒇 𝟎 =𝟏 and using a step size of Δ𝒙 = 𝟎. 𝟓, and approximate 𝒚 𝟐 using 4 steps.

EXAMPLE 2

Use Euler’s Method to approximate the particular solution of the differential equation 𝒚′ = 𝟎. 𝟓𝒙 (𝟑 − 𝒚) passing through the point 𝒇 𝟎 = 𝟏 and using a step size of 𝚫𝒙 = 𝟎. 𝟒 through 𝟓 subintervals. Fill out the table and solve for 𝒇 𝟓 .

8/7/2018 12:51 AM §6.1A: Euler's Method 17

𝒏 𝟎 𝟏 𝟐 𝟑 𝟒 𝟓

𝒙𝒏 𝟎

𝒚𝒏 𝟏

𝒏 𝟎 𝟏 𝟐 𝟑 𝟒 𝟓

𝒙𝒏 𝟎 𝟎. 𝟒 𝟎. 𝟖 𝟏. 𝟐 𝟏. 𝟔 𝟐. 𝟎

𝒚𝒏 𝟏 𝟏 𝟏. 𝟏𝟔 𝟏. 𝟒𝟓𝟒 𝟏. 𝟖𝟐𝟓 𝟐. 𝟐𝟎𝟏

5 2.201y

YOUR TURN

Use Euler’s Method to approximate the particular solution of the differential equation 𝒚′ = 𝒙 − 𝒚 passing through the point 𝒇 𝟎 = 𝟏and using a step size of 𝚫𝒙 = 𝟎. 𝟏 through 𝟖 subintervals and solve for 𝒇 𝟖 .

8/7/2018 12:51 AM §6.1A: Euler's Method 18

𝒏 𝟎 𝟏 𝟐 𝟑 𝟒 𝟓 𝟔 𝟕 𝟖

𝒙𝒏

𝒚𝒏

𝒏 𝟎 𝟏 𝟐 𝟑 𝟒 𝟓 𝟔 𝟕 𝟖

𝒙𝒏 𝟎 𝟎. 𝟏 𝟎. 𝟐 𝟎. 𝟑 𝟎. 𝟒 𝟎. 𝟓 𝟎. 𝟔 𝟎. 𝟕 𝟎. 𝟖

𝒚𝒏 𝟏 . 𝟗 . 𝟖𝟐 . 𝟕𝟓𝟖 . 𝟕𝟏𝟐 . 𝟔𝟖𝟎𝟖 . 𝟔𝟔𝟐𝟕𝟐 . 𝟔𝟓𝟔𝟒𝟒 . 𝟔𝟔𝟐𝟕𝟐

8 0.6627f

AP MULTIPLE CHOICE PRACTICE QUESTION 1 (NON-CALCULATOR)

The function 𝒇 is twice differentiable for 𝒙 > 𝟎 with 𝒇 𝟏 = 𝟏𝟓 and 𝒇′′ 𝟏 = 𝟐𝟎. Values of 𝒇′, the derivative of 𝒇, are given for selected values of 𝒙 in the table above.

(a) Write an equation for the line tangent to the graph of 𝒇 at 𝒙 = 𝟏. Use this line to approximate 𝒇 𝟏. 𝟒 .

(b) Use Euler’s method, starting at 𝒙 = 𝟏 with two steps of equal size, to approximate 𝒇 𝟏. 𝟒 . Show the computations that lead to your answer.

8/7/2018 12:51 AM §6.1A: Euler's Method 19

AP MULTIPLE CHOICE PRACTICE QUESTION 1 (NON-CALCULATOR)

The function 𝒇 is twice differentiable for 𝒙 > 𝟎 with 𝒇 𝟏 = 𝟏𝟓 and 𝒇′′ 𝟏 = 𝟐𝟎. Values of 𝒇′, the derivative of 𝒇, are given for selected values of 𝒙 in the table above.

(a) Write an equation for the line tangent to the graph of 𝒇 at 𝒙 = 𝟏. Use this line to approximate 𝒇 𝟏. 𝟒 .

8/7/2018 12:51 AM §6.1A: Euler's Method 20

: 1 15, ' 1 8Given f f

1 1y y m x x

15 8 1y x

15 8 1.4 1y

15 8 0.4y

1.4 18.2f

AP MULTIPLE CHOICE PRACTICE QUESTION 1 (NON-CALCULATOR)

The function 𝒇 is twice differentiable for 𝒙 > 𝟎 with 𝒇 𝟏 = 𝟏𝟓 and 𝒇′′ 𝟏 = 𝟐𝟎. Values of 𝒇′, the derivative of 𝒇, are given for selected values of 𝒙 in the table above.

(b) Use Euler’s method, starting at 𝒙 = 𝟏 with two steps of equal size, to approximate 𝒇 𝟏. 𝟒 . Show the computations that lead to your answer.

8/7/2018 12:51 AM §6.1A: Euler's Method 21

1

dyy y x

dx 15 ' 1 1.2 1y f

15 8 1.2 1y

15 8 0.2y

1.2 16.6f

16.6 ' 1.2 1.4 1.2y f

16.6 12 1.4 1.2y

16.6 12 0.2y

1.4 19f

ASSIGNMENT

Worksheet

8/7/2018 12:51 AM §6.1A: Euler's Method 22