Mathematical Physics I — DU B.Sc. (Prog.) Physical Science courses pertaining to Electronics Previous Year Question Papers

(a) Use chain rule to evaluate a if y sin(8) and y = x et. (b) Determine if (1 + 2xy')dx + 3x'y2 dy is an exact differential. (c) Show that ? is an integrating factor of differential equation -ydx + xdy = 0. (a) Evaluate I sin (mx) sin(nx) dx. (e) For the differential equation, x'(x-1)y"- 29x-1)y' + 3xy = 0, locate and classify its singular points) on the x-axis. (1) Given 5(2) = Vt, evaluate (g) Solve the partial differential equation by separation of variable method. (3×6=18) 2. (a) Determine the Fourier series of a periodic function f(x) with period 2t given by (b) Given f(x) = x for 0 < x < L, and f(x + 2L) = f(x), write the Fourier series representation of f(x) if it is an even function. (9,9) 3. (a) Find the dimensions x, y, z of a closed rectangular box of surface area S such that its volume is maximum. (b) Show that p(x) = ex is an integrating factor for the differential equation sin(y) dx + cos(y) dy and hence solve the differential equation. (9,9) 4. (a) Show that z=0 is a regular singular point of the differential equation (b) Use the method of Frobenius to find the series solution of Legendre's 4722 + 2ª7+y=0. differential equation d'y dx + n(n + 1)y = 0 5. (a) Evaluate S", P(x)P, (x)dx, P2(x) and P,(x) being the Legendre polynomials of degree 2 and 3 respectively. (b) Show that the beta function B(x, y) = B(y, x). Evaluate the following integral in terms of gamma functions and compute the value. 6. (a) Show that u(x, t) = sin(x) (vt) is a solution to the one-dimensional wave equation (b) The temperature T(x, y) on each of the edges of a rectangular plate is maintained as Determine the steady state temperature distribution T(x, y) in the plate by solving Laplace equation ax=+ ду2 = 0, 0<x<a, 0<y<b for f(x) = To x (a-x), I, being a constant. (6,12)