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**Unformatted text preview: **Algebraic Properties of Solutions 2nd-Order Linear Homogeneous We will begin our study of second-order differential equations with the linear homogeneous case, that is we will consider differential equations of the form y 00 + p ( x ) y + q ( x ) y = 0 . Since this equation is second-order, an IVP will have two initial conditions and be of the form y 00 + p ( x ) y + q ( x ) y = 0; y ( x ) = y , y ( x ) = y . Because these equations are linear, we have a very nice existence/uniqueness theorem: Theorem (Existence/Uniqueness) If p ( x ) and q ( x ) are continuous on the interval ( a,b ) , then there exists a unique solution to the IVP y 00 + p ( x ) y + q ( x ) y = 0; y ( x ) = y , y ( x ) = y . on the interval ( a,b ) . Remark: In light of this theorem we will henceforth assume p ( x ) and q ( x ) are continuous. Linear Operators We have seen the idea of an operator L in the context of the Picard Iterates, as a "machine" that assigns to each function f ( x ) in its domain, a new function L [ f ( x )] . That is f ( x ) L 7-→ L [ f ( x )] ....

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