MATHEMATICS-II (Ordinary Differential Equations and Complex Variables)

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

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