BE Computer Engineering (IOE, TU) Engineering Mathematics III (IOE, SH 501) Question Paper 2079

(a) State and prove the first shifting theorem for Laplace transforms. Hence evaluate $\mathcal{L}\{e^{-2t}(3\cos 4t - 2\sin 4t)\}$. [6]

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

[6]

(a) Obtain the Fourier series expansion of the function

and deduce that $\dfrac{\pi}{4} = 1 - \dfrac{1}{3} + \dfrac{1}{5} - \dfrac{1}{7} + \cdots$ [8]

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

A laterally insulated metal rod of length $L$ has its ends maintained at zero temperature. The temperature $u(x,t)$ satisfies the one-dimensional heat equation

(a) Using the method of separation of variables, derive the general solution satisfying the boundary conditions $u(0,t)=0$ and $u(L,t)=0$. [10]

(b) Hence find the temperature distribution if the initial temperature is $u(x,0) = \sin\dfrac{\pi x}{L} + 3\sin\dfrac{3\pi x}{L}$. [6]

(a) State Green's theorem in the plane. Using it, evaluate $\oint_C (3x^2 - 8y^2)\,dx + (4y - 6xy)\,dy$, where $C$ is the boundary of the region bounded by $y = \sqrt{x}$ and $y = x^2$. [7]

(b) Evaluate the surface integral $\iint_S \mathbf{F}\cdot \hat{n}\,dS$ where $\mathbf{F} = 4x\,\hat{i} - 2y^2\,\hat{j} + z^2\,\hat{k}$ and $S$ is the surface bounding the region $x^2 + y^2 = 4$, $z=0$, $z=3$, using the divergence theorem. [5]

(a) Find the Laplace transform of the periodic square wave function of period $2a$ defined by $f(t) = k$ for $0 < t < a$ and $f(t) = -k$ for $a < t < 2a$. [4]

(b) Using the convolution theorem, find $\mathcal{L}^{-1}\left\{\dfrac{1}{s^2(s^2+1)}\right\}$. [4]

Find the series solution of the ordinary differential equation

about the regular singular point $x=0$ using the Frobenius method. Determine the indicial equation and obtain at least one of the two linearly independent solutions.

(a) Prove the recurrence relation $\dfrac{d}{dx}\left[x^n J_n(x)\right] = x^n J_{n-1}(x)$ for Bessel functions. [4]

(b) Using Rodrigues' formula, obtain the Legendre polynomials $P_2(x)$ and $P_3(x)$, and verify the orthogonality relation $\displaystyle\int_{-1}^{1} P_2(x)P_3(x)\,dx = 0$. [4]

(a) State the Cauchy-Riemann equations. Show that the function $f(z) = \bar{z}$ is nowhere analytic. [3]

(b) Show that $u(x,y) = x^3 - 3xy^2 + 3x^2 - 3y^2 + 1$ is harmonic and find its harmonic conjugate $v(x,y)$ such that $f(z) = u + iv$ is analytic. [5]

(a) State Cauchy's residue theorem. [2]

(b) Using the residue theorem, evaluate $\displaystyle\oint_C \frac{z+1}{z^2 - 2z}\,dz$, where $C$ is the circle $|z| = 3$ described in the positive sense. [6]

(a) Form the partial differential equation by eliminating the arbitrary function $f$ from $z = f(x^2 + y^2 + z^2)$. [3]

(b) Solve the linear partial differential equation $x(y - z)\,p + y(z - x)\,q = z(x - y)$ using Lagrange's method, where $p = \partial z/\partial x$ and $q = \partial z/\partial y$. [5]

(a) Show that the vector field $\mathbf{F} = (2xy + z^3)\hat{i} + x^2\hat{j} + 3xz^2\hat{k}$ is conservative, and find a scalar potential $\phi$ such that $\mathbf{F} = \nabla\phi$. [4]

(b) Verify Stokes' theorem for $\mathbf{F} = y\,\hat{i} + z\,\hat{j} + x\,\hat{k}$ over the upper half of the surface of the sphere $x^2 + y^2 + z^2 = 1$ bounded by its projection on the $xy$-plane. [4]