IGNOU PHE-05 - Mathematical Methods in Physics-II

1. a) Solve the following ordinary differential equations: i) y’ = (y -1) cot x ii) y” + 2y’ + y = e- 2x b) Solve the initial value problem: 2. A parachutist is falling with a speed of 100 ms-1 when his parachute opens. If the air resistance is (Mv2)/ 25 where M is the total mass of the man and his parachute, find the speed of the man as a function of time t after the parachute opens. Take g = 10 ms-1. 3. A 1 kg mass is attached to a spring of spring constant 12 Nm-1 and the system is immersed in a medium which exerts a damping force equal to 8 times the instantaneous velocity of the mass. Determine the position of the mass as a function of time if it is released from rest at a point 0.5 m below its equilibrium position. 4. Determine the roots of the indicial equation around the point x = 0 for the following ODE: 5. a) Show that: is a solution of the one-dimensional heat equation. b) Determine all the first and second order partial derivatives for the function: u (x, y)= x2 sin y + y2 cos x 6. Write the following partial differential equation in spherical polar coordinates and reduce it to a set of three ODEs by the method of separation of variables: ∇2 f + k2 f = 0 7. Obtain the Fourier series for the following odd function which has a period of 2π: f(x)= x sin 2 x for - π < x < π 8. Solve the heat conduction problem: Given that T (0,t) = T (5,t) = 0 and T (x,0) = 3sin(πx) - 2sin(2πx) + sin(5πx)