View by Category

The presentation will start after a short

(15 second) video ad from one of our sponsors.

Hot tip: Video ads won’t appear to registered users who are logged in. And it’s free to register and free to log in!

(15 second) video ad from one of our sponsors.

Hot tip: Video ads won’t appear to registered users who are logged in. And it’s free to register and free to log in!

Loading...

PPT – Differential Equations PowerPoint presentation | free to view - id: f4111-ZDNhZ

The Adobe Flash plugin is needed to view this content

About This Presentation

Write a Comment

User Comments (0)

Transcript and Presenter's Notes

Differential Equations

- First-order differential equations
- Concepts
- Solution of a differential equation
- Separable equations
- Homogeneous and inhomogeneous equations
- Second-order differential equations
- Concepts
- Homogeneous linear equations
- The general solution
- The particle in a 1-d box
- Special functions and the power series method
- Partial differential equations
- General solutions and an example

Differential equations are equations that contain

derivatives

- Examples
- dx/dt kx 1st order rate process
- dx/dt k(a-x)(b-x) 2nd order rate process
- d2h/dt2 -g Body falling by gravity
- (-h2/2m)d2?/dx2 ½kx2? e?
- Wave equation for harmonic oscillator.

General Solutions first order

- General function f(x,y,dy/dx,d2y/dx2,) 0
- By solving, or integrating, the differential

equation our aim is to find the function(s) y(x)

that satisfy the equation. - dy/dx 2x? See whiteboard

Separable equations

- Consider dy/dx f(x)/g(y)
- We can write this as
- g(y)dy f(x)dx
- This is called a separable differential equation.
- See whiteboard

First order linear equations

- these can be written
- dy/dx p(x)y r(x)
- if r(x) 0 this is a homogeneous equation.
- if not this is an inhomogeneous equation.
- First order linear homogeneous differential

equations are separable and are therefore easily

soluble. - First order linear inhomogeneous differential

equations require just a little more work. - See whiteboard

Second-order differential equations. Constant

coefficients.

- d2y/dx2p(x)dy/dx q(x)y r(x)
- p(x) and q(x) are called the coefficients of the

equation. We start with the cases where p(x) and

r(x) are constants.

Homogenous equations

- d2y/dx2 a.dy/dx b.y 0
- Always possible to find a solution of the form

e?x where ? is some suitable constant. - Example, see whiteboard.

General solution

- See whiteboard.
- The equation y?2y 0
- Solutions y(x) c1ei?x c2e-i?x
- or
- y(x) d1 cos ?x d2 sin ?x

Particle in a 1-d box

- Schödinger equation
- (-h2/2m)d2?(x)/dx2 V(x)?(x) E?(x)
- V(x) is zero inside box. Infinite outside box.
- So, inside the box
- (-h2/2m)d2?(x)/dx2 E?(x)
- with ?(0) ?(l)0

Particle in a 1-d box

- (-h2/2m)d2?(x)/dx2 E?(x)
- Set ?2 (2mE)/h2
- Eqn becomes
- d2?/dx2 ?2 ? 0
- The equation y?2y 0
- Solutions y(x) c1ei?x c2e-i?x
- or
- y(x) d1 cos ?x d2 sin ?x

Particle in a 1-d box

- d2?/dx2 ?2 ? 0
- has solution ?(x) d1 cos ?x d2 sin ?x
- Boundary conditions
- First ?(0) d1 cos 0 d2 sin 0 0
- So d1 0.
- Second ?(l) d2 sin ?l 0, which is only true

when ?l is some multiple of ?, i.e. ?l n ? (n

is some ve integral number, quantization!)

Particle in a 1-d box

- So, solution
- ?(x) d2 sin (n?x/l), n 1, 2, 3, ..
- For each solution (each n) there is a given E.
- How can we get d2?

Special functions

- Solving the equation y?2y 0, is useful in

many problems HO, PIB, PIR - There are other common diff. equations that pop

up frequently enough for us to consider them

special. e.g. - Legendre (1-x2)y-2xyl(l1)y 0
- Hermite y-2xy2ny 0
- One way of solving these is with a power series.

Simple illustration of the power series method

- See whiteboard.

About PowerShow.com

PowerShow.com is a leading presentation/slideshow sharing website. Whether your application is business, how-to, education, medicine, school, church, sales, marketing, online training or just for fun, PowerShow.com is a great resource. And, best of all, most of its cool features are free and easy to use.

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

Recommended

«

/ »

Page of

«

/ »

Promoted Presentations

Related Presentations

Page of

Page of

CrystalGraphics Sales Tel: (800) 394-0700 x 1 or Send an email

Home About Us Terms and Conditions Privacy Policy Contact Us Send Us Feedback

Copyright 2017 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

Copyright 2017 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

The PowerPoint PPT presentation: "Differential Equations" is the property of its rightful owner.

Do you have PowerPoint slides to share? If so, share your PPT presentation slides online with PowerShow.com. It's FREE!

Committed to assisting Unt University and other schools with their online training by sharing educational presentations for free