BE Computer Engineering (Pokhara University) Calculus II (PU, MTH 210) Question Paper 2079

(a) If $u = f(x, y)$ where $x = r\cos\theta$ and $y = r\sin\theta$, show that

(7)

(b) Examine the function $f(x, y) = x^3 + y^3 - 3axy$ (where $a > 0$) for maxima, minima and saddle points. (7)

(a) Evaluate the double integral $\displaystyle \int_0^1 \int_{x^2}^{\sqrt{x}} xy \, dy \, dx$ and sketch the region of integration. (6)

(b) By changing the order of integration, evaluate $\displaystyle \int_0^a \int_y^a \frac{x}{x^2 + y^2} \, dx \, dy$. (4)

(c) Using triple integration, find the volume of the solid bounded by the cylinder $x^2 + y^2 = 4$ and the planes $z = 0$ and $z = 3 - x$. (4)

(a) Solve the differential equation $\dfrac{d^2 y}{dx^2} - 4\dfrac{dy}{dx} + 4y = e^{2x} + \sin 2x$. (7)

(b) Solve the equation $(D^2 + 1)y = \sec x$ using the method of variation of parameters, where $D \equiv \dfrac{d}{dx}$. (5)

(a) State the convolution theorem for Laplace transforms and use it to find $L^{-1}\left\{\dfrac{1}{(s^2 + a^2)^2}\right\}$. (6)

(b) Using the Laplace transform method, solve the initial value problem $y'' + 4y = \sin t$, given that $y(0) = 0$ and $y'(0) = 0$. (6)

If $u = x + y + z$, $uv = y + z$ and $uvw = z$, find the Jacobian $\dfrac{\partial(x, y, z)}{\partial(u, v, w)}$.

(a) Find the directional derivative of $\phi = x^2 yz + 4xz^2$ at the point $(1, -2, -1)$ in the direction of the vector $2\hat{i} - \hat{j} - 2\hat{k}$. (4)

(b) Show that the vector field $\vec{F} = (y^2 \cos x + z^3)\hat{i} + (2y \sin x - 4)\hat{j} + (3xz^2 + 2)\hat{k}$ is irrotational. (3)

Verify Green's theorem in the plane for $\displaystyle \oint_C \left[(3x^2 - 8y^2)\,dx + (4y - 6xy)\,dy\right]$, where $C$ is the boundary of the region bounded by the lines $x = 0$, $y = 0$ and $x + y = 1$.

Use Stoke's theorem to evaluate $\displaystyle \oint_C \vec{F} \cdot d\vec{r}$ where $\vec{F} = (2x - y)\hat{i} - yz^2\,\hat{j} - y^2 z\,\hat{k}$ and $C$ is the boundary of the upper half of the sphere $x^2 + y^2 + z^2 = 1$ traversed in the positive direction.

(a) Solve the first-order differential equation $\dfrac{dy}{dx} + y\cot x = \cos x$. (3)

(b) Solve the exact differential equation $(2xy + y - \tan y)\,dx + (x^2 - x\tan^2 y + \sec^2 y)\,dy = 0$. (4)

Obtain the Fourier series expansion of the function $f(x) = x^2$ in the interval $-\pi < x < \pi$, and hence deduce that $\dfrac{\pi^2}{6} = \sum_{n=1}^{\infty} \dfrac{1}{n^2}$.

Find the power series solution of the differential equation $\dfrac{d^2 y}{dx^2} + x\dfrac{dy}{dx} + y = 0$ about the ordinary point $x = 0$, obtaining the recurrence relation and the first few terms of the two independent solutions.

(a) Find the Laplace transform of $t\,e^{-2t}\sin 3t$. (3)

(b) Express the function $f(t) = \begin{cases} t, & 0 < t < 2 \\ 4 - t, & 2 < t < 4 \\ 0, & t > 4 \end{cases}$ in terms of unit step functions and hence find its Laplace transform. (3)