Mathematics II (BT-202) - Important Questions

Solve $(1+y^2)dx = (\tan^{-1}y - x)dy$ as a linear equation in $x$ as function of $y$.

Solve a second-order Cauchy-Euler / Legendre homogeneous linear differential equation $x^2 y'' + x y' + y = f(x)$ by reduction to constant coefficients.

Solve $(e^y+1)\cos x\,dx + e^y\sin x\,dy = 0$.

Solve second-order linear ODE with constant coefficients with sinusoidal RHS $\sin ax$ / $\cos ax$.

Solve second-order linear ODE with constant coefficients with exponential plus constant RHS.

Solve the homogeneous simultaneous linear differential equations $\frac{dx}{dt}-7x+y=0$ and $\frac{dy}{dt}-2x-5y=0$.

Solve $dy/dx = \cos(x+y)+\sin(x+y)$ by substitution $u=x+y$.

Solve second-order linear ODE $y''+y' = (1+e^x)^{-1}$ requiring variation of parameters.

Solve the non-homogeneous simultaneous system $\frac{dx}{dt}-y=e^t$, $\frac{dy}{dt}+x=\sin t$ with initial conditions $x(0)=1$, $y(0)=0$.

Solve $\left(1+e^{x/y}\right)dx + e^{x/y}\left(1-x/y\right)dy = 0$.

Solve $\cos x\,dy = y(\sin x - y)\,dx$ using Bernoulli's method.

Solve the linear equation $(1+y^2)+(x-e^{-\tan^{-1}y})dy/dx = 0$.

Solve $(D^2+a^2)y = \tan ax$ (including special cases $a=2,3$) by the method of variation of parameters.

Show that $J_{1/2}(x) = \sqrt{\frac{2}{\pi x}} \sin x$.

Solve in series Legendre's differential equation $(1-x^2)y''-2xy'+2y=0$.

Solve $(D^2+1)y = x$ by the method of variation of parameters.

Show that $J_n(-x) = (-1)^n J_n(x)$ when $n$ is a positive or negative integer.

Solve the differential equation $x(1-x)y''+2(1-2x)y'-2y=0$ using the Frobenius method.

Solve a linear partial differential equation with constant coefficients with sinusoidal right-hand side.

Solve the first-order linear PDE $Pp+Qq=R$ by Lagrange's method

Solve a linear partial differential equation with constant coefficients with polynomial right-hand side.

Form a first-order partial differential equation by eliminating a single arbitrary function from a relation of the form $z = g(x,y) + f(u(x,y))$.

Solve by Charpit's method $(p^2+q^2)y = qz$

Solve by Charpit's method $px+qy=pq$

Construct a partial differential equation from an implicit relation $f(u(x,y,z), v(x,y,z)) = 0$ involving an arbitrary function of two arguments.

Solve $x^2p^2+y^2q^2=z^2$

Solve a linear partial differential equation with constant coefficients with exponential right-hand side.

Find the poles and the residues at each pole for a given complex function involving higher-order or simple poles.

Determine the constant parameter $K$ (or $P$) such that the given function of $x$ and $y$ is analytic using Cauchy-Riemann equations.

Use Cauchy integral formula for derivatives to evaluate $\int_C \frac{f(z)}{(z-a)^n}dz$ where $f$ is analytic inside $C$.

Evaluate the complex line integral $\int_C f(x,y)\,dz$ along a specified path from $z=0$ to $z=1+i$ (or $(0,0)$ to $(1,1)$).

Show that $u(x,y)$ of exponential-trigonometric form is harmonic and find its harmonic conjugate / construct the corresponding analytic function $f(z)=u+iv$.

Show that $u = e^{-x}(x\sin y - y\cos y)$ is harmonic.

Determine $p$ so that $f(z) = \frac{1}{2}\log(x^2+y^2) + i\tan^{-1}(px/y)$ is analytic.

Prove that an analytic function with constant modulus is constant.

Use Cauchy integral formula for simple poles to evaluate $\oint_C \frac{f(z)}{(z-a)(z-b)}dz$ with poles inside $C$.

Using complex integration method, solve: $\int_0^{2\pi} \frac{\cos 4\theta}{5 + 4\cos\theta} d\theta$

Write a short note on: (i) Cauchy-Riemann equations (ii) Stokes theorem.

Prove that $\left[\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right]|f(z)|^2 = 4|f'(z)|^2$ for analytic $f$.

Prove that $r^n \bar{r}$ is irrotational, i.e. $\text{curl}(r^n \bar{r}) = \bar{0}$.

Find the directional derivative of a scalar function $f(x,y,z)$ at a given point in a given direction (including direction given by a vector or by the tangent to a parametric curve).

Verify Green's theorem for a given line integral over a plane region $C$.

Verify Stokes theorem for $\bar{F} = (x^2-y^2)\bar{i}+2xy\bar{j}$ over the box bounded by $x=0,x=a,y=0,y=b$.

Find the workdone by the force $\bar{F} = z\bar{i} + x\bar{j} + y\bar{k}$, when it moves a particle along the arc of the curve $\bar{r} = \cos t\,\bar{i} + \sin t\,\bar{j} - t\bar{k}$ from $t = 0$ to $t = 2\pi$.

Verify Gauss divergence theorem for $\mathbf{F}$ over a cube bounded by coordinate planes.

Show that $\frac{\vec{r}}{r^3}$, i.e. $r^n \vec{r}$ with $n=-3$, is solenoidal (divergence zero).

Prove that $\nabla^2 f(r) = f''(r) + \frac{2}{r}f'(r)$.

Using Green's theorem, find the area of the region in the first quadrant bounded by given curves.

Find the angle between the surfaces $x^{2} + y^{2} + z^{2} = 9$ and $Z = x^{2} + y^{2} - 3$ at the point $(2, -1, 2)$.

Show that the vector $(x^2-yz)\hat{i}+(y^2-zx)\hat{j}+(z^2-xy)\hat{k}$ is irrotational and find its scalar potential.