Loading in 5 sec....

Section 6.1: Euler’s MethodPowerPoint Presentation

Section 6.1: Euler’s Method

- 101 Views
- Uploaded on

Section 6.1: Euler’s Method. Local Linearity and Differential Equations.

Download Presentation
## PowerPoint Slideshow about ' Section 6.1: Euler’s Method' - tamyra

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

Local Linearity and Differential Equations

Local Linearity can be used to numerically approximate an equation without knowing the particular solution. Consider the differential equation if . (The blue curve below is the particular solution. Assume this is unknown.)

Not a good approximation. Consider smaller pieces.

Slope at (2,0):

Tangent line at (2,0):

Tangent line approximation at x=2.2:

Local Linearity and Differential Equations

Local Linearity can be used to numerically approximate an equation without knowing the particular solution. Consider the differential equation if . (The blue curve below is the particular solution. Assume this is unknown.)

Slope at (2.2,0.4):

“Tangent line” at (2.2,0.4):

“Tangent line” approximation at x=2.4:

Local Linearity and Differential Equations

Local Linearity can be used to numerically approximate an equation without knowing the particular solution. Consider the differential equation if . (The blue curve below is the particular solution. Assume this is unknown.)

Slope at (2.4,0.92):

“Tangent line” at (2.4,0.92):

“Tangent line” approximation at x=2.6:

Local Linearity and Differential Equations Local Linearity can be used to numerically approximate an equation without knowing the particular solution. Consider the differential equation if . (The blue curve below is the particular solution. Assume this is unknown.)

Slope at (2.6,1.584):

“Tangent line” at (2.6,1.584):

“Tangent line” approximation at x=2.8:

Local Linearity and Differential Equations Local Linearity can be used to numerically approximate an equation without knowing the particular solution. Consider the differential equation if . (The blue curve below is the particular solution. Assume this is unknown.)

Slope at (2.8,2.421):

We can generalize this process.

“Tangent line” at (2.8,2.421):

“Tangent line” approximation at x=3:

Derivative

Previous y

Change in x

Next y

Euler’s Method

Numerically approximate values for the solution of the initial-value problem , , with step size , at , are

Change in x

Next Approximate Solution

Previous Approximate Solution

Value of Differential Equation at Previous Point

Example

If and if when , use Euler’s Method with five equal steps to approximate when .

Euler’s Method

5

Download Presentation

Connecting to Server..