1. State whether the following statements are true or false. Give a sh

Question

1. State whether the following statements are true or false. Give a short proof or a counterexample in support of your answer:

a) A real-valued function of three variables which is continuous everywhere is differentiable.

b) The function $f\left ( x \right )=ln\left ( \frac{x+y}{x} \right )$ is not homogeneous function.

c) The cylindrical coordinates of the point whose spherical coordinates is $\left ( 8,\frac{\pi }{6},\frac{\pi }{2} \right )\: is\: \! \! \left ( 8,\frac{\pi }{6},0 \right ).$

d) The unique solution y(x) of an ordinary differential equation:

$\frac{dy}{dx}=$ $\frac{dy}{dx}= \left\{\begin{matrix} 0, for \,\, \, x< 0 & \\ 1, for\, \, x\geq 0 & \end{matrix}\right.$

exists $\forall$ $x\in \mathbb{R}.$

e) The differential equation $\left [ 1+\left ( y \right )^{2} \right ]^{\frac{5}{3}}=y$ is a second order differential equation of degree 3.

f) Differential equation: $cos\: x\frac{d^{2}y}{dx^{2}}+\frac{dy}{dx}+xy^{2}=0$

in $]$ $0,\pi$ $[$ is a linear, homogeneous equation.

g) The total differential equation corresponding to the family of surfaces , $x^{3}z+x^{2}y=c,$ where c is a parameter is . $3x^{2}dz+x\left ( ydx+xdy \right )=0.$

h) Differential equation:

$5x^{2}y^{2}z^{2}=2px^{2}y^{2}+5qx^{2}y^{3}+2pz^{2}+9x^{2}y^{2}$

is a semi- linear partial differential equation of first order.

i) The differential equation:

$v\frac{du}{dy}=e^{2v}+uv-u$

has an integrating factor v exp(−v) .

j) The simultaneous differential equation of simple harmonic motion of a particle in phase –plane is: $\frac{dx}{y}=\frac{dy}{-w^{2}x}=y_{0}$

02 Mar 2024

Answer :

Word Count : 689

Let's go through the statements one by one: a) A real-valued function of three variables which is continuous everywhere is differentiable. False. A function being continuous does not necessarily mean it is differentiable. A common counterexample is the function $ f(x, y, z) = |x| $, which is continuous everywhere but not differentiable at $ x = 0 $. --- b) The function $ f(x, y) = \ln\left( \frac{x + y}{x} \right) $ is not a homogeneous function. True. A function is homogeneous of degree $ n $ if for any scalar $ t $, $ f(tx, ty) = t^n f(x, y) $. For this function: $ f(tx, ty) = \ln\left( \frac{tx + ty}{tx} \right) = \ln\left( \frac{x + y}{x} \right) = f(x, y) $. Thus, it is not homogeneous because it does not satisfy the condition _______ ___ ___ __________ ________ ________ __________ __________ _____ ____.
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