BE Computer Engineering (IOE, TU) Engineering Mathematics III (IOE, SH 501) Question Paper 2078

(a) Define the Laplace transform of a function $f(t)$ and state the conditions of existence. Find the Laplace transform of $f(t) = e^{-2t}\,t\cos 3t$ using the appropriate shifting and differentiation theorems. [6]

(b) Using the Laplace transform method, solve the initial value problem

[6]

(a) Obtain the Fourier series expansion of the periodic function defined by

with period $2\pi$, and hence deduce that $\displaystyle \sum_{n=1}^{\infty} \frac{1}{(2n-1)^2} = \frac{\pi^2}{8}$. [8]

(b) Find the half-range cosine series for $f(x) = x$ in the interval $0 < x < 2$. [4]

A laterally insulated thin rod of length $L$ has its ends at $x=0$ and $x=L$ maintained at zero temperature. The initial temperature distribution is $u(x,0)=f(x)$.

(a) Using the method of separation of variables, derive the solution of the one-dimensional heat equation $\dfrac{\partial u}{\partial t} = c^2 \dfrac{\partial^2 u}{\partial x^2}$ subject to the given boundary and initial conditions. [8]

(b) Hence write down the temperature distribution when $f(x) = 3\sin\dfrac{\pi x}{L} - 5\sin\dfrac{4\pi x}{L}$. [4]

(a) Using the Frobenius (power series) method, find the series solution of the differential equation

about the regular singular point $x=0$, obtaining the indicial equation and the recurrence relation. [8]

(b) Show that the Bessel equation of order $n$, $x^2 y'' + x y' + (x^2 - n^2)y = 0$, reduces to the elementary form whose solution for $n=\tfrac12$ can be expressed in terms of $\dfrac{\sin x}{\sqrt{x}}$. [4]

(a) State and prove the convolution theorem for Laplace transforms, and use it to evaluate $\mathcal{L}^{-1}\left\{\dfrac{1}{(s^2+a^2)^2}\right\}$. [5]

(b) Find the Laplace transform of the full-wave rectified sine wave $f(t)=|\sin t|$, which is periodic with period $\pi$. [3]

(a) Evaluate the line integral $\displaystyle \oint_C \left[(3x^2 - 8y^2)\,dx + (4y - 6xy)\,dy\right]$ where $C$ is the boundary of the region bounded by $y=\sqrt{x}$ and $y=x^2$, using Green's theorem. [5]

(b) Determine whether the vector field $\vec{F} = (2xy + z^3)\hat{i} + x^2\hat{j} + 3xz^2\hat{k}$ is conservative, and if so find its scalar potential. [3]

(a) Verify the Divergence (Gauss) theorem for $\vec{F} = x^2\hat{i} + y^2\hat{j} + z^2\hat{k}$ taken over the surface of the cube bounded by $x=0,\,x=1,\,y=0,\,y=1,\,z=0,\,z=1$. [5]

(b) State Stokes' theorem and explain its physical significance. [3]

(a) State Rodrigues' formula for the Legendre polynomials $P_n(x)$ and use it to obtain $P_2(x)$ and $P_3(x)$. [4]

(b) Prove the orthogonality property $\displaystyle \int_{-1}^{1} P_m(x)\,P_n(x)\,dx = 0$ for $m \neq n$. [4]

(a) Derive the Cauchy-Riemann equations and use them to show that $f(z)=z^2$ is analytic everywhere, finding $f'(z)$. [4]

(b) Given the harmonic function $u(x,y) = x^3 - 3xy^2$, find the conjugate harmonic function $v(x,y)$ and express the corresponding analytic function $f(z)=u+iv$ in terms of $z$. [4]

(a) Form the partial differential equation by eliminating the arbitrary functions from $z = f(x+ct) + g(x-ct)$. [3]

(b) Using d'Alembert's method, obtain the solution of the one-dimensional wave equation $\dfrac{\partial^2 y}{\partial t^2} = c^2 \dfrac{\partial^2 y}{\partial x^2}$ for an infinite string with initial displacement $y(x,0)=\phi(x)$ and initial velocity $\dfrac{\partial y}{\partial t}(x,0)=0$. [5]

(a) Find the complex (exponential) form of the Fourier series of $f(x)=e^{x}$ on the interval $-\pi < x < \pi$. [5]

(b) State Parseval's identity for Fourier series and explain its significance in terms of average power. [3]