►PROBLEMS, SECTION 11.2.3.4.5.6.If u = x²/(x² + y2), find Ju/0x, du/dy.If st", find əs/at, 08/0u.If z = ln √u² + v² +w², find əz/au, dz/dv, dz/dw.For w=a³y³ - 2xy + 6, find a2 w/0x² and 2 w/y² at the points whereƏw/dx = Əw/y = 0.For w=8a¹ + y - 2xy², find ² w/8² and 0w/y² at the points whereдw/дх= дw/ду = 0.For u = e cos y,(a) verify that 0²u/dxdy = ²u/əyəx;(b) verify that 0²u/0x² +0²u/0y² = 0.

Answered: If u = x²/(x² + y²), find ou/ox, du/dy.

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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Related questions

Question

►PROBLEMS, SECTION 1
1.
2.
3.
4.
5.
6.
If u = x²/(x² + y2), find Ju/0x, du/dy.
If st", find əs/at, 08/0u.
If z = ln √u² + v² +w², find əz/au, dz/dv, dz/dw.
For w=a³y³ - 2xy + 6, find a2 w/0x² and 2 w/y² at the points where
Əw/dx = Əw/y = 0.
For w=8a¹ + y - 2xy², find ² w/8² and 0w/y² at the points where
дw/дх= дw/ду = 0.
For u = e cos y,
(a) verify that 0²u/dxdy = ²u/əyəx;
(b) verify that 0²u/0x² +0²u/0y² = 0.

Expert Solution

Step 1

# As per the guidelines we are entitled to solve one question at a time, please resubmit the other question if you wish to get it answered.

Given $u = \frac{x^{2}}{x^{2} + y^{2}}$

To find the value of $\frac{\partial u}{\partial x}$ and $\frac{\partial u}{\partial y}$ proceed as follows.

$\begin{array}{rcl} \frac{\partial u}{\partial x} & = & \frac{\partial (\frac{x^{2}}{x^{2} + y^{2}})}{\partial x} \\ = & \frac{(x^{2} + y^{2}) (2 x) - x^{2} (2 x)}{{(x^{2} + y^{2})}^{2}} \\ = & \frac{2 x^{3} + 2 x y^{2} - 2 x^{3}}{{(x^{2} + y^{2})}^{2}} \\ = & \frac{2 x y^{2}}{{(x^{2} + y^{2})}^{2}} \end{array}$