MATHEMATICS-II (Ordinary Differential Equations and Complex Variables) - B.Tech 2nd Semester Exam., 2019

Find the directional derivative of \(\varphi(x,y,z)=x^{2}yz+4~xz^{2}\) at \((1,-2,-1)\) in the direction \(2i-j-2k\)

Evaluate \(\nabla.[r\nabla(1/r^{3})].\)

What is the degree of the differential equation \(\left(\frac{d^{3}y}{dx^{3}}\right)^{2/3}+\left(\frac{d^{3}y}{dx^{3}}\right)^{3/2}=0\)?

Find the general solution of the differential equation \(x(x^{2}+3y^{2})dx+y(y^{2}+3x^{2})dy=0\)

Evaluate the integral \(\int_{C}\frac{(e^{z}+\sin~\pi z)dz}{(z-1)(z+1)(z+4)},\) \(C:|z|=2\)

Evaluate the integral \(\int_{C}\frac{dz}{(z^{2}+4z+3)^{2}}\) \(C:|z|=4\)

Define the pole-type singularity with an example.

Find the bilinear transformation that maps \(z_{1}=\infty,\) \(z_{2}=i\) and \(z_{3}=0\) into the points \(w_{1}=0,\) \(w_{2}=i\) and \(w_{3}=\infty.\)

If \( a{<}b,\) then evaluate the integral \(\int_{a}^{b}|(x-a)+(x-b)|dx \)

Evaluate the integral \(\int_{0}^{\infty}\int_{x}^{\infty}\frac{e^{-y}}{y}dy~dx\)

Evaluate the integral \(\int_{0}^{a}\int_{y}^{a}\frac{x}{(x^{2}+y)^{2}}dy~dx\)

Find the mass of a plate in the form of a quadrant of an ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) whose density per unit area is given by \(\rho=kxy.\)

Evaluate \(\int_{C}F\cdot dr,\) where \(F=(3x^{2}+6y)i-14yzj+20xz^{2}k\) from \((0,0,0)\) to \((1,1,1)\) along the path \(x=t,\) \(y=t^{2}\) and \(z=t^{3}\)

Evaluate \(\int_{C}F\cdot dr\) along the straight line joining \((0,0,0)\) to \((1,1,1)\)

Solve the differential equation \((x^{2}+y^{2}+x)dx-(2x^{2}+2y^{2}-y)dy=0\)

Solve the differential equation \(y=2px+y^{2}p^{3}\)

State and prove Rodrigues' formula.

Show that \(2nJ_{n}(x)=x[J_{n+1}(x)+J_{n-1}(x)]\)

Find the series solution of the differential equation \(x^{2}\frac{d^{2}y}{dx^{2}}+6~x\frac{dy}{dx}+(x^{2}+6)y=0\)

State and prove the sufficient condition for a function \(w=f(z)\) to be analytic.

Find an analytic function \(f(z)\) such that \(\operatorname{Re}\{f^{\prime}(z)\}=3x^{2}-4y-3y^{2}\) and \(f(1+i)=0\)

Discuss the nature of the singularities for \(\left(\frac{1-\cosh~z}{z^{3}}\right).\) Also determine the order of the pole and corresponding residue if it exists.

Find what regions of the w-plane correspond by the transformation \(w=\left(\frac{z-i}{z+i}\right)\) to the interior of a circle of centre \(z=-i.\)

Evaluate \(\oint_{C}\frac{\sin^{2}z}{z(z-1)(2z+5)}dz,\) \(C:|z-1|+|z+1|=3\)

Evaluate \(\int_{0}^{\infty}\frac{\sin(mx)}{x(x^{2}+a^{2})}dx\)