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

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