Chapter3Section2

# Chapter3Section2 - 3.2 General Solutions of Linear...

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3.2: General Solutions of Linear Equations It all extends .... Definition A n th order linear differential equation can be written in the form y H n L + p 1 H x L y I n - 1 M + + p n H x L y = f H x L , where the functions p 1 H x L , ..., p n H x L , f H x L are continuous on an I . If f H x L 0 for all x in I , then we say it is h o m o g e n e o u s . Other- wise, it is n o n - h o m o g e n e o u s . Theorem - Principle of Superposition for Homogeneous Equations Let y 1 , ..., y n be solutions to y H n L + p 1 H x L y I n - 1 M + + p n H x L y = 0. Then y = c 1 y 1 + c 2 y 2 + + c n y n is also a solution.

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Theorem - Existence and Uniqueness for Linear Equations If p 1 H x L , ..., p n H x L , f H x L are continuous on an open interval I and a Ε I . Then, given any b 0 , b 1 , ..., b n - 1 and a ˛ I , y H n L + p 1 H x L y I n - 1 M + + p n H x L y = f H x L , has a unique solution on I that satisfies the initial conditions y H a L = b 0 , y ' H a L = b 1 , ..., y I n - 1 M H a L = b n - 1 . Definition Functions f 1 , ..., f n are linearly dependent on an interval I if there exists constants, c 1 , ..., c n
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