1 / 8

Existence of a Unique Solution

Existence of a Unique Solution. Let the coefficient functions and g(x) be continuous on an interval I and let the leading coefficient function not equal 0 for every x in I. If x is any point in this interval, then a solution y(x) of the initial-value problem exists on the interval and is unique.

allen-kerr
Download Presentation

Existence of a Unique Solution

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. Existence of a Unique Solution Let the coefficient functions and g(x) be continuous on an interval I and let the leading coefficient function not equal 0 for every x in I. If x is any point in this interval, then a solution y(x) of the initial-value problem exists on the interval and is unique.

  2. Homogeneous EquationsNonhomogeneous Equations

  3. Superposition Principle—Homogeneous Equations Let be solutions of the homogeneous nth-order differential equation on an interval I. Then the linear combination where the coefficients are arbitrary constants, is also a solution on the interval.

  4. Corollaries • A constant multiple of a solution of a homogeneous linear differential equation is also a solution. • A homogeneous linear differential equation always possesses the trivial solution .

  5. Linear Dependence/Independence A set of functions is said to be linearly dependent on an interval I if there exist constants , not all zero, such that for every x in the interval. If the set of functions is not linearly dependent on the interval, it is said to be linearly independent.

  6. Criterion for Linearly Independent Solutions Let be n solutions of the homogeneous linear nth-order differential equation on an interval I. Then the set of solutions is linearly independent on I if and only if for every x in the interval.

  7. Fundamental Set of Solutions Let set of n linearly independent solutions of the homogeneous linear nth-order differential equation on an interval I is said to be a fundamental set of solutions on the interval. There exists a fundamental set of solutions for the homogeneous linear nth-order DE on an interval I.

  8. General Solution-Homogeneous Equation Let be a fundamental set of solutions of the homogeneous linear nth-order differential equation on an interval I. Then the general solution of the equation on the interval is where the coefficients are arbitrary constants.

More Related