BE Computer Engineering (Pokhara University) Calculus II (PU, MTH 210) Question Paper 2078

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

(b) State Euler's theorem on homogeneous functions and use it to verify the theorem for $u = \tan^{-1}\!\left(\dfrac{x^3 + y^3}{x - y}\right)$, hence evaluate $x\dfrac{\partial u}{\partial x} + y\dfrac{\partial u}{\partial y}$.

(c) Examine the function $f(x,y) = x^3 + y^3 - 3axy$ for maxima, minima and saddle points.

(a) Evaluate the double integral $\displaystyle \iint_R (x^2 + y^2)\,dx\,dy$ over the region $R$ bounded by the lines $y = x$, $y = 0$ and $x = a$ by first sketching the region of integration.

(b) Change the order of integration in $\displaystyle \int_{0}^{1}\int_{x^2}^{2-x} xy\,dy\,dx$ and hence evaluate it.

(c) Using a triple integral, find the volume of the region in the first octant bounded by the plane $\dfrac{x}{a} + \dfrac{y}{b} + \dfrac{z}{c} = 1$.

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

(b) Verify Stokes' theorem for the vector field $\vec{F} = (2x - y)\hat{i} - yz^2\hat{j} - y^2 z\,\hat{k}$ 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.

(a) Define the Laplace transform of a function $f(t)$ and find $\mathcal{L}\{t\,e^{-2t}\sin 3t\}$ from first principles using the standard shifting and multiplication-by-$t$ properties.

(b) Using Laplace transforms, solve the initial value problem

(c) State the convolution theorem and use it to find the inverse Laplace transform of $\dfrac{1}{(s^2 + a^2)^2}$.

If $u = x + y + z$, $uv = y + z$ and $uvw = z$, find the Jacobian $\dfrac{\partial(x,y,z)}{\partial(u,v,w)}$ and state whether the transformation is invertible at the point $(u,v,w) = (1, 1, 1)$.

For the scalar field $\phi = x^2 yz + 4xz^2$, find $\nabla\phi$ at the point $(1,-2,-1)$. Also, for $\vec{F} = xz^3\hat{i} - 2x^2 yz\,\hat{j} + 2yz^4\hat{k}$, compute $\nabla\cdot\vec{F}$ and $\nabla\times\vec{F}$ at the same point.

Solve the differential equation $(x^2 + y^2 + x)\,dx + xy\,dy = 0$ by identifying a suitable integrating factor, and obtain its general solution.

Find the general solution of the second-order linear differential equation

by determining both the complementary function and a particular integral.

Using the method of partial fractions, find the inverse Laplace transform

Obtain the Fourier series expansion of the function $f(x) = x^2$ in the interval $-\pi < x < \pi$, and hence deduce that

Find the half-range sine series for the function $f(x) = \pi - x$ in the interval $0 < x < \pi$.

Obtain the series solution about the ordinary point $x = 0$ of the differential equation

finding the recurrence relation and the first few non-zero terms of the two linearly independent solutions.