1 / 12

Chapter 8: Ordinary Differential Equations I. General A linear ODE is of the form:

Ch. 8- Ordinary Differential Equations > General. Chapter 8: Ordinary Differential Equations I. General A linear ODE is of the form: An n th order ODE has a solution containing n arbitrary constants ex: ex:. Ch. 8- Ordinary Differential Equations > General. Three really common ODE’s:

danhunt
Télécharger la présentation

Chapter 8: Ordinary Differential Equations I. General A linear ODE is of the form:

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. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Ch. 8- Ordinary Differential Equations > General • Chapter 8: Ordinary Differential Equations • I. General • A linear ODE is of the form: • An nth order ODE has a solution containing n arbitrary constants • ex: • ex:

  2. Ch. 8- Ordinary Differential Equations > General • Three really common ODE’s: • 1) • 2) • 3)

  3. Ch. 8- Ordinary Differential Equations > General • How do we solve for the constants? • → In general, any constant works. • → But many problems have additional constants (boundary conditions) and in this case, the particular solution involves specific values of the constants that satisfy the boundary condition. • ex: for t<0, the switch is open and the capacitor is uncharged. • at t=0, shut switch

  4. Ch. 8- Ordinary Differential Equations > Separable Ordinary Differential Equations II. Separable Ordinary Differential Equations A separable ODE is one in which you can separate all y-terms on the left hand side of the equation and all the x-terms on the right hand side of the equation. ex: xy’=y We can solve separable ordinary differential equations by separating the variables and then just integrating both sides

  5. Ch. 8- Ordinary Differential Equations > Separable Ordinary Differential Equations ex: xy’=y subject to the boundary condition y=3 when x=2

  6. Ch. 8- Ordinary Differential Equations > Separable Ordinary Differential Equations ex: Rate at which bacteria grow in culture is proportional to the present. Say there are no bacteria at t=0. subject to boundary condition N(t=0)=No

  7. Ch. 8- Ordinary Differential Equations > Separable Ordinary Differential Equations ex: Schroedinger’s Equation: solve for the wave function if V(x,t) is only a function of x, e.g. V(x), then schroedinger’s equation is separable.

  8. Ch. 8- Ordinary Differential Equations > Linear First-Order ODEs III. Linear First-Order Ordinary Differential Equations Definition: a linear first-order ordinary differential equation can be written in the form: y’+Py=Q where P and Q are functions of x the solution to this is: Check:

  9. Ch. 8- Ordinary Differential Equations > Linear First-Order ODEs ex:

  10. Ch. 8- Ordinary Differential Equations > Second-Order Linear Homogeneous Equation IV. Second Order Linear Homogeneous Equation A second order linear homogeneous equation has the form: where a2, a1, a0 are constants To solve such an equation: let

  11. Ch. 8- Ordinary Differential Equations > Second-Order Linear Homogeneous Equation ex: y’’+y’-2y=0

  12. ex: Harmonic Oscillator

More Related