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 - 1.4 Separable Differential Equations - Problems - Page 43: 8

Answer

\[\sin x\cos y=C\]

Work Step by Step

$\large\frac{dy}{dx}=\large\frac{\cos(x-y)}{\sin x\sin y}-\small1$ ___(1) $\large\frac{dy}{dx}=\frac{\cos x\cos y+\sin x\sin y}{\sin x\sin y}-1$ $\large\frac{dy}{dx}=\frac{1}{\tan x\tan y}$ Separating variables, $\;\;\tan y\; dy=\cot x\;dx$ Integrating, $K+\int\tan y\; dy=\int\cot x\;dx$ $K$ is constant of integration $K-\ln|\cos y|=\ln |\sin x|$ $K=\ln|\sin x\cos y|$ $\ln C=\ln|\sin x\cos y|$ Where $K=\ln C$ $C=\sin x\cos y$ Solution of (1) is $\;\;\sin x\cos y=C$
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