**Question 1:** Show that, if $ u(x, y) = x^2 - y^2 + 2x + 1 $ and $ v(x, y) = 2xy + 2y $, then \[\frac{1}{4(x^2 + y^2 + 2x + 1)} \left( \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} \right) = \frac{\partial^2 f}{\partial u^2} + \frac{\partial^2 f}{\partial v^2}.\]

Answered: Question 1. Show that, if u(r, y) =…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Given,

$u (x, y) = x^{2} - y^{2} + 2 x + 1, v (x, y) = 2 x y + 2 y$

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$\frac{1}{4 (x^{2} + y^{2} + 2 x + 1)} (\frac{\partial^{2} f}{\partial x^{2}} + \frac{\partial^{2} f}{\partial y^{2}}) = \frac{\partial^{2} f}{\partial u^{2}} + \frac{\partial^{2} f}{\partial v^{2}}$

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