Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 3 - Section 3.11 - Hyperbolic Functions - 3.11 Exercises - Page 265: 55

Answer

(a) $y'' = m^2y$ (b) $y = 2~sinh~3x-4~cosh~3x$

Work Step by Step

(a) $y = A~sinh~mx+B~cosh~mx$ $y' = mA~cosh~mx+mB~sinh~mx$ $y'' = m^2A~sinh~mx+m^2B~cosh~mx = m^2y$ (b) $y'' = m^2y$ If $y'' = 9y,$ then $m^2=9$ and $m = 3$ If $y(0) = -4$: $A~sinh~m(0)+B~cosh~m(0) = -4$ $0+B~(1) = -4$ $B = -4$ If $y'(0) = 6$: $mA~cosh~m(0)+mB~sinh~m(0) = 6$ $mA(1)+B~(0) = 6$ $mA = 6$ $A = \frac{6}{3}$ $A = 2$ Therefore: $y = 2~sinh~3x-4~cosh~3x$
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