Applied Mathematics 2023

Forces of magnitudes P, 2P, 3P, 4P act respectively along the sides AB, BC, CD, DA of a square ABCD, of side a and forces each of magnitude (8√2) P act along the diagonals BD, AC. Find the magnitude of the resultant force and the distance of its line of action from A.

A uniform rod AB of length a and weight W is freely hinged to a vertical wall at A and is maintained in equilibrium by a light string of length a fastened to B and to a point C at a distance b vertically above A. Prove that the reaction at the hinge A is and find the tension in the string.

Use Runge-Kutta method of order two to solve the following differential equation at x=1.2 by taking h=0.1 , y(1) = 1.

Find the first and second derivatives of f(x) at x = 3 from the following data using Newton's forward difference interpolation formula

X      3      3.5     4      4.5     5      5.5
f(x)  4.1023  5.1047  8.1971  9.1096  4.1122  6.1148

Find the angle between the surfaces $x^2 + y^2 + z^2 = 9$ and $z = x^2 + y^2 – 3$ at the point (2, -1, 2).

Show that

Find the total work done in a moving particle in a force field given by $F = 3xy \hat{i} − 5 z \hat{j} + 10 x \hat{k}$ along the curve $x = t^2 + 1, y = 2t^2, z = t^3$ from $t = 1$ to $t = 2$.

A particle P moves in a plane in such a way that at any time t, its distance from a fixed point O is $r = a t + b t^2$ and the line connecting O and P makes an angle $\theta = ct^2$ with a fixed line OA. Find the radial and transverse components of the velocity and acceleration of the particle at $t = 1$.

Solve the following Bernouli's equation

x 
dy
dx + y = 
1
y^2

Solve the following differential equation

x 
dy
dx = (x sinx – y) dx

Find the general solution of the higher order differential equation

y''' + 8y' = −6 x² + 9x + 2

Find solution of $4y'' + y = 0$ in the form of power series in x.

Solve the following differential equation by variation of parameters

y" – 4y' + 4y = (x + 1)e^{2x}

Find real root of the equation $2x – 3 \sin(x) – 5 = 0$ up to 4 decimal places by secant method.

Solve the following system of equations by Guass Seidel method. Perform only five iterations.

8x₁ - x₂ - x₃ = 6
x₁ + 6x₂ + x₃ = 8
x₁-x₂ + 5x₃ = 5

Expand $f(x) = \sin x, 0 < x < \pi$, in a Fourier cosine series.

Use the method of separation of variables to find the solution of the following boundary value problem

∂²u
∂x² + 
∂²u
∂y² = 0, 0 ≤ x ≤ a, 0≤ y ≤b
ux(0,y) = 0, ux(a, y) = 0,
u_{y}(x,b) = 0, u(x, 0) = f(x).