Applied Mathematics 2025

(a) (i) Prove that $\nabla r^n = nr^{n-2} \vec{r}$, where $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$.

(ii) $\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \times \vec{b}) \times \vec{c}$, then prove that $\vec{a}$ and $\vec{c}$ are parallel.

(b) Find the area of the region that is enclosed between the curves $y = x^2$ and $y = x+6$.

(a) Find the tangential and normal components of acceleration of a point describing the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with uniform speed, when the particle is at $\left(0, b\right)$.

(b) Find the solution of initial value problem by separation of variables

$\sqrt{1-y^2} dx - \sqrt{1-x^2} dy = 0$, $y(0) = \frac{\sqrt{3}}{2}$

(a) Find the general solution of the given differential equation by variation of parameters.

$3y" - 6y'+6y = e^x \sec x$

(b) Find the power series solution of $(x^2+1)y" + xy' - y = 0$

(a) Forces $2\vec{BC}$, $\vec{CA}$, $\vec{BA}$ act along the sides of a triangle ABC. Show that their resultant is $6\vec{DE}$. Where D bisects BC and E is a point on CE such that $CE = \frac{1}{3} CA$.

(b) Find the center of mass of the surface generated by the revolution of the arc of the parabola, lying between the vertex and the latus rectum, about the x-axis.

(a) Obtain the Fourier series over the indicated interval for the given function.

$f(x) = \begin{cases} 3\pi + 2x, & - \pi < x < 0 \\ \pi+2x, & 0 < x < \pi \end{cases}$

(b) Solve the boundary value problem

$\nabla^2 u = u_{xx}+u_{yy} = 0$, $0

$u(0, y) = 0$, $u(a, y) = 0$, $0\le y\le b$

$u(x, 0) = 0$, $u(x,b) = f(x)$, $0 \le x \le a$.

(a) Use Newton's Raphson method to find the solution accurate to within $10^{-4}$ (corrected upto four decimal places) for the given problem.

$x-\cos x = 0$, $[0, \pi/2]$.

(b) Solve the system of linear equations using Gauss Seidel method (with three digit rounding arithmetic)

$3x_1 +4x_2-x_3 = 8$

$5x_1 +3x_2 + 2x_3 = 17$

$-x_1 + x_2 – 3x_3 = -8$

(a) Use Euler's method to approximate the solution of the initial value problem.

$y'=1+y/x$, $1 \le x \le 2$, $y(1) = 2$, with $h=0.25$

(b) Using Green's theorem, evaluate $\oint_{C} \vec{F} \cdot d\vec{r}$ counter clock wise around the boundary curve $C$ of the region $R$, where $\vec{F} = \left[ \frac{1}{2}xy^3, \frac{1}{2}x^2y^2 \right]$ is the rectangle with vertices $(0, 0), (3, 0), (3, 2), (0, 2)$.

(a) Evaluate the Integral $\int_{1}^{3} \frac{1}{x} dx$, Using Trapezoidal Rule for five points (corrected upto two decimal places).

(b) Find the D'Alembert solution of the wave equation $u_{xx} = \frac{1}{c^2} u_{tt}$ subject to the Cauchy Initial conditions $u(x, 0) = f(x)$, $u_t(x, 0) = g(x)$.