(i) \(\mathrm{u}=\tan ^{-1}\left(\frac{x^{3}+y^{3}}{x-y}\right)\)
u is not a homogeneous function
Let \(\tan \mathrm{u}=\left(\frac{x^{3}+y^{3}}{x-y}\right)=\mathrm{f}(\mathrm{x}, \mathrm{y})\)
\(\mathrm{f}(\lambda \mathrm{x}, \lambda \mathrm{y})=\frac{\lambda^{3} x^{3}+\lambda^{3} y^{3}}{\lambda x-\lambda y}\)
\(=\frac{\lambda^{3}\left(x^{3}+y^{3}\right)}{\lambda(x-y)}\)
f is a homogeneous function of degree 2
By Euler's Theorem,
\( \mathrm{x} \frac{\partial f}{\partial x}+\mathrm{y} \frac{\partial f}{\partial y} =2 \mathrm{f} \)
\(\mathrm{x} \frac{\partial}{\partial x} \tan \mathrm{u}+\mathrm{y} \frac{\partial}{\partial y} \tan u =2 \tan \mathrm{u} \)
\(\mathrm{x}\ \mathrm{sec}\ \mathrm{u} \frac{\partial u}{\partial x}+\mathrm{y} \mathrm{} \sec ^{2} \mathrm{u} \frac{\partial u}{\partial y} =2 \tan u \)
\(\mathrm{x} \frac{\partial u}{\partial x}+\mathrm{y} \frac{\partial u}{\partial y} =\frac{2 \tan u}{\sec ^{2} u} \)
\( =\frac{2 \sin u}{\cos u} \times \cos ^{2} u \)
\(\mathrm{x} \frac{\partial u}{\partial x}+\mathrm{y} \frac{\partial u}{\partial y} =\sin 2 \mathrm{u} \)
(ii) \( \mathrm{u}(\mathrm{x}, \mathrm{y}) =\mathrm{xy}^{2} \sin (x / y) \)
\(\mathrm{u}(\lambda \mathrm{x}, \lambda \mathrm{y}) =\lambda \mathrm{x} \lambda^{2} \mathrm{y}^{2} \sin (\lambda x / \lambda y) \)
\( =\lambda^{3} \mathrm{xy} \mathrm{y}^{2} \sin (x / y) \)
\(\therefore\) u is a homogeneous function of degree 3.
\(\therefore \mathrm{x} \frac{\partial u}{\partial x}+\mathrm{y} \frac{\partial u}{\partial y}=3 \mathrm{u}\)
(iii) \( \mathrm{u} =\left(x^{2}+y^{2}\right)^{1 / 2} \)
\(\mathbf{u}(\lambda \mathrm{x}, \lambda \mathrm{y}) =\left(\lambda^{2} x^{2}+\lambda^{2} y^{2}\right)^{1 / 2} \)
\( =\lambda\left(x^{2}+y^{2}\right)\)
u is a homogeneous function of degree 1. By Euler's Theorem,
\(x \frac{\partial u}{\partial x}+y \frac{\partial u}{\partial y}=u\)
(iv) \( \mathrm{u} =e^{(x)} \sin (x / y)+e^{(\% / x)} \cos (y / x) \)
\(\mathrm{u}(\lambda \mathrm{x}, \lambda \mathrm{y}) =e^{\lambda / / y} \sin \lambda x / / y+e^{2 / / x} \cos \pi y / \lambda x \)
\( =e^{(y)} \sin (x / y)+e^{(\% / x)} \cos (y / x) \)
u is a homogbneous function of degree 0 By Euler's Theorem,
\(\mathbf{x} \frac{\partial u}{\partial x}+\mathrm{y} \frac{\partial u}{\partial y}=0\)

U = (x - y)(y - z)(z - x)
log U = log (x- y) + log (y - z) + log (z - x)
\( \frac{1}{U} \frac{\partial U}{\partial x}=\frac{1}{x-y}-\frac{1}{z-x} \)
\( \frac{1}{U} \frac{\partial U}{\partial y}=\frac{1}{y-z}-\frac{1}{x-y} \)
\( \frac{1}{U} \frac{\partial U}{\partial z}=-\frac{1}{y-z}+\frac{1}{z-x} \)
\( \frac{1}{U}\left(\frac{\partial U}{\partial x}+\frac{\partial U}{\partial y}+\frac{\partial U}{\partial z}\right)= \frac{1}{x-y}-\frac{1}{z-x}+\frac{1}{y-z} -\frac{1}{x-y}-\frac{1}{y-z}+\frac{1}{z-x} \)
= 0
\(\frac{\partial U}{\partial x}+\frac{\partial U}{\partial y}+\frac{\partial U}{\partial z}=\mathrm{U} \times 0\)
= 0

u = log (tan x + tan y + tan z)
\( \frac{\partial u}{\partial x} =\frac{1}{\tan x+\tan y+\tan z} \times \sec ^{2} x \)
\(\sin 2 x \frac{\partial u}{\partial x} =\frac{2 \sin x \cos x \times \frac{1}{\cos ^{2} x}}{\tan x+\tan y+\tan z} \)
\( =\frac{2 \tan x}{\tan x+\tan y+\tan z} \)
Similarly,
\( \sin 2 \mathrm{y} \frac{\partial u}{\partial y}=\frac{2 \tan y}{\tan x+\tan y+\tan z} \)
\( \sin 2 z \frac{\partial u}{\partial z}=\frac{2 \tan z}{\tan x+\tan y+\tan z} \)
\( \sum \sin 2 x \frac{\partial u}{\partial x}=\frac{2(\tan x+\tan y+\tan z)}{\tan x+\tan y+\tan z}=2\)

u(x, y) = \(x^{4}+y^{3}+3 x^{2} y^{2}+3 x^{2} y\) 
\( \frac{\partial u}{\partial x}=4 x^{3}+6 x^{2}+6 x y \)
\( \frac{\partial^{2} u}{\partial y \partial x}=12 x y+6 x\)
\( \frac{\partial u}{\partial y} =3 y^{2}+6 x^{2} y+3 x^{2} \)
\(\frac{\partial^{2} u}{\partial x \partial y} =12 x y+6 x \)
\( \frac{\partial^{2} u}{\partial x \partial y} =\frac{\partial^{2} u}{\partial y \partial x}\)

−sint+2cost+2t

0

\(\frac { 5 }{ 18 } \)

\(\frac { 1 }{ 4 } \)

\(\frac { \partial w }{ \partial v } =4u\left( { u }^{ 2 }+{ v }^{ 2 } \right) ;\frac { \partial w }{ \partial v } =4v\left( { u }^{ 2 }+{ v }^{ 2 } \right) \)

\(\frac { 3 }{ 2 } \)

\(\frac { 1 }{ 4 } \)

0.677

0.2867

2

\(\frac { \partial w }{ \partial r } =\frac { 2 }{ r } ,\frac { \partial w }{ \partial \theta } =0\)