Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 5 - The Integral - Chapter Review Exercises - Page 278: 36


$$y=\frac{1}{3 \pi }\sin (3\pi t)-\frac{1}{4\pi }\cos (4\pi t) - \frac{\pi}{8}$$

Work Step by Step

Given$$ \frac{d y}{d t}=\cos (3\pi t)+\sin (4\pi t) , \quad y(1/3)= 0 $$ Since :\begin{align*} y&= \int (\cos (3\pi t)+\sin (4\pi t) ) dt\\ &=\frac{1}{3 \pi }\sin (3\pi t)-\frac{1}{4\pi }\cos (4\pi t) +c \end{align*} at $t=\frac{1}{3},\ \ \ y= 0 $, then $c= -\pi/8$, hence: $$y=\frac{1}{3 \pi }\sin (3\pi t)-\frac{1}{4\pi }\cos (4\pi t) - \frac{\pi}{8}$$
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