3.2 - Math 334 Lecture #10 § 3.2: Linear Second Order...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 334 Lecture #10 § 3.2: Linear Second Order ODE’s, Part I The IVP for the general linear second order ODE is y 00 + p ( t ) y + q ( t ) y = g ( t ) , y ( t ) = y , y ( t ) = y . Geometrically, the IVP asks for a solution of the ODE which passes through the point ( t , y ) with slope y . Existence and Uniqueness Theorem for IVP. If the functions p ( t ), q ( t ), and g ( t ) are continuous on an open interval I that contains t , then there exists a unique solution y = φ ( t ), t ∈ I , of the IVP. Example. On what open interval is the solution of the following IVP defined? (1- t 2 ) y 00- y = exp {- t 2 } , y (0) = 0 , y (0) = 1 . To correctly identify the functions p , q , and g the ODE needs to be rewritten: y 00- 1 1- t 2 y = exp {- t 2 } 1- t 2 . The largest open interval containing t = 0 on which p , q , and g are continuous is I = (- 1 , 1), which is the interval of definition of the solution of the IVP. [Knowing that the solution of the IVP exists does NOT tell us how to find it. We begin a journey to develop the tools needed to solve an IVP, at least in principle.] The Differential Operator Notation. Let L denote the differential operator which takes a twice differentiable function...
View Full Document

This note was uploaded on 03/08/2011 for the course MATH 334 taught by Professor Smith during the Spring '11 term at Vanderbilt.

Page1 / 3

3.2 - Math 334 Lecture #10 § 3.2: Linear Second Order...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online