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 - 1.8 Change of Variables - Problems - Page 79: 3


$f(x,y)$ is homogeneous of degree zero. $$f(v)=\frac{ \sin \frac{1}{v}- v\cos v}{ v}$$

Work Step by Step

Given $$f(x, y)=\frac{ x \sin\frac{x}{y}- y\cos\frac{y}{x}}{ y}$$ Let \begin{aligned} f (t x, ty)&=\frac{ t x \sin\frac{tx}{ty}- ty\cos\frac{ty}{tx}}{ ty}\\ &=\frac{ t x \sin\frac{x}{y}- ty\cos\frac{y}{x}}{ ty}\\ &=\frac{ t( x \sin\frac{x}{y}- y\cos\frac{y}{x})}{ ty}\\ &=\frac{ x \sin\frac{x}{y}- y\cos\frac{y}{x}}{ y}\\ &=f(x,t) \end{aligned} Thus, $f(x,y)$ is homogeneous of degree zero. Transforming the given function \begin{aligned}f(x, y)&=\frac{ \frac{x}{x} \sin\frac{x}{y}- \frac{y}{x}\cos\frac{y}{x}}{ \frac{y}{x}}\\ &=\frac{ \sin\frac{x}{y}- \frac{y}{x}\cos\frac{y}{x}}{ \frac{y}{x}}\\ \\ \text{Put} \ \ \ v=\frac{y}{x} \ \ \ \text{so, we get}\\ f(x,y)&=\frac{ \sin \frac{1}{v}- v\cos v}{ v}\\ \end{aligned}
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