Integral Equations and Calculus of Variations — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

Integral Equations and Calculus of variations Bachelor of Science Glonours Course) Mathematics 1. (a) Find resolvent kernel for the following integral equation (b) Solve by successive approximations : (c) Solve the integral equation 2. (a) Solve convolution equation dy{x3 = cos¢x) - {x ~ t} cos€x - t} ¢tt) dt. (b) Solve the following integral equation of first kind by first reducing it to integral equation of second kind a;x-t dyct}at --f cx3, f an = 0- (c) Solve die integral equation i/ox(#_£)2¢(t)d£=cos(#)-1+i. (a) Show that the function p(x) = xe-X is a solution of the Fredholm-type integral equation ap(#) -4 e-(r+f)p(t)d£ = (x - 1)G-r. (b) Using Fredholm determinates, find the resolvent kernel of the kemel K(x,t)=sin(#)-sin(€); Osa:S271, Osts27r. (c) Find the iterated kernels K](x, f) and K2(x, f) if K(x, t) = ex cos(t), a = 0, b - 71:. 4. (a) Solve the integral equation ap(#) - 1 K`x,t)ap(t)di--x (b) Solve the given Fredholm integral equation ap(#) - fi sin(# - I) p(£)tzf = cos(x) where ^ is a parameter. (c) Find the characteristic numbers and eigenfunctions of the integral equation ap(I) - A. K¢, £) -cos.(x)sin(€), i.f os#S£ cos(t)sin¢), if £SxS7r (a) Among all the curvesjoining two givenpoints (xo, yo) and (x], y]), find the one which generates the surface of minimum area when rotated about the x -axis. (b) Among all the curves lying on the sphere x2 + y2 + z2 = a2 and passing through two given points (xo, yo, zo) and (x], y], z]), find the one which has the least length. (c) Find the shortest distance of point P(0,0,4) from line x = y = z. (a) Find the path on which a particle in the absence of friction, will slide from one point to another in the shortest time under the action of gravity. (b) Among all curves of length 1 in the upper half-plane passing through the points (-a, 0) and (a, 0), find the one which together with the interval [-a, a] encloses the largest area. (c) Find the shortest distance of point P(0,2,3) from line x/2 = y/3 = z/4.