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 : 715

Let's analyze each statement one by one. ### a) A real-valued function of three variables which is continuous everywhere is differentiable. False. A function that is continuous everywhere does not necessarily have to be differentiable everywhere. A classic counterexample is the function $ f(x, y, z) = |x| $, which is continuous everywhere, but not differentiable at $ x = 0 $. ### b) The function $ f(x) = \ln \left( \frac{x+y}{x} \right) $ is not a homogeneous function. True. A homogeneous function satisfies the property $ f(tx, ty) = t^n f(x, y) $ for some degree $ n $. In this case, $ f(x, y) = \ln \left( 1 + \frac{y}{x} \right) $, and scaling $ x $ and $ y $ by $ t $ doesn't satisfy this property, so the function is not homogeneous. ### c) The cylindrical coordinates of the point whose spherical coordinates are $ \left( 8, \frac{\pi}{6}, \frac{\pi}{2} \right) $ is $ \left( 8, \frac{\pi}{6}, 0 \right) $. False. The spherical coordinates $ (r, \theta, \phi) = (8, \frac{\pi}{6}, \frac{\pi}{2}) $ correspond __________ _____ ________ ____ ________ _________ _________ _______ ________ ___ ______ ______.
__________ _________ ____ _____ __________.
_____ ______ ____ _______ __________.
___ __________ ______ ___ ____ ______ _________ ______ ____ ___ ______.
_________ _______ ______ _________ __________ ______ ____ ______ ______.
______ __________ _______ ______ _______ _________ ______ __________ ______ _________ __________.
___ _____ _________ __________ __________ ________ __________ ______ _________ _____ ____.
_____ ____ ______ ____ _____ _________ ____ ________ _________ ________ _____ ___.
___ ______ __________ _______ _____ ____ ____ __________ __________.
_________ ______ ______ __________ _________ _______ ________.
___ ___ __________ ______ ________ _____.
_____ ______ _____ _________ ______.
_________ _________ _________ ______ ____ ______ ________ _____ ________ ______ _______ ______.
__________ _________ ______ ______ ______ ____ _________ _______ ________.
_______ ________ __________ ___ ________ _____ _______ _______ ___ ___.
_______ ______ _____ _________ ___ __________ _____ _______ __________ _____ ______ ______.
____ ________ _________ _________ _______ ________ _____ ______ ________.
_______ ______ _____ ______ _______ _______ ___ ____ ____ ___ ______ _______.
_________ __________ _____ _________ _____ ________ ________ ______.
_____ _________ ____ ______ ______ ______ ______ _____.
____ ______ _________ __________ ________ ___ _____ ___ ______.
_______ ____ __________ __________ ____.
_________ ____ _____ __________ ___ ______ _________ ______ _________ _________.
______ ________ _______ _________ ____ ________ ___ ____ _________.
________ _______ _________ ________ ______ __________ _____ ___ _____ ______.
_________ _________ ____ ___ ____ _________ _________ _______.
_________ ____ ______ ____ _______ ______ ___.
_______ __________ ___ __________ _____ __________ ___ ___ ___.
__________ _________ ______ ____ _______ _____ _________.
___ __________ _____ _________ ________ _____ ________ ________ _______ _______.
________ ______ ________ ___ _______ _______ __________ ________ ____.
__________ _____ __________ ____ ________ _________ _________ ________ ____.
________ __________ ___ __________ ________ ________ ____ ______ _____ _________.
______ ____ _______ _________ ______ ____ _______ ____ ___ ___ _______ __________.
________ __________ ____ _______ _________ _______ _______.
________ ____ _______ __________ __________ ______ __________ ____ __________ __________.
_________ ______ _________ __________ ___ ___ _______ __________ ________ _____ ______ _______.
_______ _______ _______ _______ ______ ______.
____ ________ ________ ________ ___ ______ _________ _________ __________ _______ _____.
_________ _______ _____ ______ ________ ____ ________ __________.
________ ___ _____ ____ ______ ____ ________ ______ ________ __________.
_______ ________ ____ ________ _________ __________.
________ ____ ____ _______ _________ __________ _______.
__________ _____ ___ _________ _______ __________ _______ ____.
__________ ___ ________ ____ ______ ______ _____ __________.
___ ____ ________ ____ ______ ___ _______ ________ ____ _________ ___.
____ ______ ______ ___ ________ ________ _________ __________ ____ ______ ___.
________ ______ __________ _____ _______ ____ _____ _______ ____ ______.
______ _______ _________ ____ _________ ________ ___ ____ ________ ___.
___ _________ __________ ________ ___ _____.
_______ ___ ___ ________ _______ ________ __________ ________.
_______ _________ _____ __________ ____ ________ ____ __________ _____ _____ ____ ____.
____ _____ ___ ____ ___.
____ ______ _____ __________ ______ ___.
_________ _______ ________ ___ _______ _______ __________ ____ __________ _________.
_________ _____ ___ _______ ______.
__________ ___ _________ __________ ______.
__________ ____ _________ _____ ______ __________ _________ __________.
_______ _________ __________ ______ _________ ______ ___ __________ __________.
_____ ________ ________ ________ ______ _____ __________ _____ __________ ______ ________ ______.
_____ ___ _______ __________ ______ _________ ____ _________ ______.
Get Full Answer on WhatsApp

IGNOU Solved Assignments Assignments Ignou Projects Project Previous Year Solved Exam Papers Notes Notes Sample Question Papers Sample Question Papers

IGNOU NEWS

Assignment Submission Last Date Extended Till 30 June 2026 Click Here★★★IGNOU June 2026 TEE Date Sheet Released Click Here★★★