Question

ques for all

Answer 1

Paper:Mathematical Physics - I
Maximum Marks: 75
Time:3 Hours
Attempt any 5 Questions
Q1)
a)By calculating the Wronskian of the functions -1, sin^2(x), cos^2(x), check whether the functions are linearly dependent or independent(4)
b)Solve the inexact equation(5)
\[y(xy+2x^2y^2)dx+x(xy-x^2y^2)dy=0\]
c)Solve the differential equation(6)
\[\frac{d^2y}{dx^2}-2\frac{dy}{dx}+y=e^x+x\]
Q2)
a) solve the differential equation:(7)
\[\frac{d^2y}{dx^2}-4y=x\sin(x)\]
b)Solve the differential equation using method of undertermined coefficients(8)
\[\frac{d^2y}{dx^2}-4\frac{dy}{dx}+4y=x^2+\cos(2x)\]
Q3)
a) Solve the differential equation(8)
\[\frac{d^2y}{dx^2}-\frac{dy}{dx}-2y=10\cos(x)\]
given y(0)=1 and y'(0)=-1
b) Solve the differential equation using method of variation of parameters(7)
\[\frac{d^2y}{dx^2}+a^2y=\sec(ax)\]
Q4) 5 parts of 3 marks each
a) Find the volume of a parallelepiped whose sides are given by
\[\vec A=2 \hat i+3 \hat j-\hat k\]\[\vec B=\hat i-\hat j-2 \hat k\]\[\vec C=-\hat i+2 \hat j+2\hat k\]
b)Calculate the Jacobian \[J(\frac{x,y,z}{u,v,w})\]of the transformation
\[x=u+w^2 \space \space ; \space \space y=u+v \space \space ; \space \space z=w^2-u\]
c) If \[\vec v=\vec w \times \vec r\]
Find whether v is solenoidal or not, where w is a constant vector and \[\vec r=x \hat i+y \hat j+z \hat k\]
d) Find\[\vec \nabla.(f(r)\vec r) \space \space \space ; \space \space \space \vec r=x \hat i+y \hat j+z \hat k \]
e)Find the directional derivative of a scalar function
\[\phi=(x^2+y+z^2)^{-\frac{1}{2}}\] at the point P(3,1,2) in the direction of the vector
\[yz \hat i+xz \hat j+xy \hat k\]
Q5.)
a) Prove that(3)
\[(\vec A \times \vec B).(\vec C \times \vec D)=(\vec A . \vec C)(\vec B. \vec D)-(\vec A.\vec D)(\vec B.\vec C)\]
b)Evaluate(6)
\[\nabla^2[\vec \nabla.(\frac{\vec r}{r^2})] \space \space \space ; \space \space \space \vec r=x \hat i+y \hat j+z \hat k\]
c)Evaluate(6)
\[\oint_\limits C (3x^2-8y^2)dx+(4y-6xy)dy\]
Where C is the boundary of the region defined by y^2=x and y=x^2
Q6)
a)Verify Stokes' Theorem for:(10)
\[\vec A=(2x-y)\hat i-yz^2 \hat j-y^2z \hat k\]
Where S is the upper half of the unit sphere and C is it's boundary
b)Using Gauss Divergence theorem, prove that (5)
\[\iiint_\limits V \vec \nabla \phi dV=\iint_\limits S \phi d \vec S\]
Where V is the volume enclosed by the surface S
Q7)
a)Derive an expression of divergence of a vector field in orthogonal curvilinear coordinates. Express it in cylindrical coordinates (6)
b)Evaluate(6)
\[\iiint_\limits V (x^2+y^2+z^2)dV\]
Where V is the volume of the sphere x^2+y^2+z^2=a^2
c)Define the Dirac Delta function and establish(3)
\[\int\limits_{-\infty}^{\infty}f'(x)\delta(x-a)dx=-f'(a)\]

@IrishBoy123 Thanks for the help...it went great...thought I'd share it anyway
@ganeshie8 @Michele_Laino

Answer 2

well done, dude👓

and it is really nice of you to come back and share.

Answer 3

which ones did you do?

Answer 4

According to the first few lines you've typed out, this is a test (paper) for which you earn credit for correctly answered problems. Please be aware that it's strictly against the rules of OpenStudy for you to ask for help with credit-bearing work or for other members of OpenStudy to provide it.

Answer 5

I did 1,3,4,5,7

Answer 6

@mathmale Nope, I'm just sharing my question paper I had for my exam yesterday.

Answer 7

Hey may i see your answer for 7c ?

Answer 8

Hmm unfortunately I only gave the definition and wrote some properties, the syllabus is really messed up...No one knows why they have put Dirac delta function...
Some properties are

\[\delta(Cx)=\frac{1}{|C|}\delta(x)\]
For some constant C
\[x\delta(x)=0\]
\[\delta(-x)=\delta(x)\]\[\int\limits_{-\infty}^{\infty}\delta(x)dx=1\]