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: 4

Answer

$f(x,y)$ is homogeneous of degree zero. $$f(x,y)=\frac{\sqrt{3 +5 V^2}}{2 +5 V}$$

Work Step by Step

Given $$f(x, y)=\frac{\sqrt{3 x^{2}+5 y^{2}}}{2 x+5 y}$$ Let \begin{aligned} f (t x, ty)&=\frac{\sqrt{3(t x)^{2}+5(t y)^{2}}}{2 t x+5 t y}\\ &=\frac{t \sqrt{3 x^{2}+5 y^{2}}}{t(2 x+5 y)}\\ &=\frac{\sqrt{3 x^{2}+5 y^{2}}}{2 x+5 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{\sqrt{3 \frac{x^{2}}{x^2}+5\frac{y^2}{ x^{2}}}}{2 \frac{x}{x}+5 \frac{y}{x}}\\ &=\frac{\sqrt{3 +5\frac{x^{2}}{y^2} }}{2 +5 \frac{y}{x}}\\ \\ \text{Put} \ \ \ V=\frac{y}{x} \ \ \ \text{so, we get}\\ f(x,y)&=\frac{\sqrt{3 +5 V^2}}{2 +5 V}\\ \end{aligned}
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.