Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 9: First-Order Differential Equations - Practice Exercises - Page 557: 1


$y=-ln [c-\dfrac{2}{5} (x-2)^{5/2}-\dfrac{4}{3} (x-2)^{3/2}]$

Work Step by Step

Re-arrange the given equation as: $y'=xe^y(\sqrt {x-2})$ Now, $\int e^{-y} dy=\int x\sqrt {x-2} dx$ Consider $a=x-2$ or, $ da=dx$ $ e^{-y} +c=\int (a+2) \sqrt a da$ or, $ e^{-y} =c-(\dfrac{2}{5}) a^{5/2}-(\dfrac{4}{3}) a^{3/2}$ This implies that $ \ln [e^{-y}] =\ln [c-(\dfrac{2}{5}) a^{5/2}-(\dfrac{4}{3}) a^{3/2}]$ Hence, $y=-ln [c-\dfrac{2}{5} (x-2)^{5/2}-\dfrac{4}{3} (x-2)^{3/2}]$
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