Differential Equations and Linear Algebra (4th Edition)

Published by Pearson
ISBN 10: 0-32196-467-5
ISBN 13: 978-0-32196-467-0

Chapter 1 - First-Order Differential Equations - 1.2 Basic Ideas and Terminology - Problems - Page 22: 41


See below.

Work Step by Step

Take derivatives of the equation. Don't forget to use chain rule. $$y(x)=c_1e^x+c_2e^{-x}$$ $$y'(x)=c_1e^x-c_2e^{-x}$$ $$y''(x)=c_1e^x+c_2e^{-x}$$ Substituting these functions into the differential equation yields $$y''-y=0$$ $$c_1e^x+c_2e^{-x}-c_1e^x-c_2e^{-x}=0$$ $$0=0$$ This statement is always true, therefore, this solution is valid always on any interval.
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