Problems in Methods of Applied Mathematics

7. Obtain an alternative integral-equation formulation of the problem described by equations (11) and (12). by first setting y* - u, and showing that y(x) = (x - ¢)u( e) d� + y�(x - a) + Yo where u(x) satisfies an integral equation of the form ______ u(x) = f [(e = x)B(x) � A(x))u(e) de + F(x). 8. Show that the application of the method of Section 3.2 to the problem y" + Ay' + By = 0, y(0) = y(l) = 0, where A and B are constants, leads to the integral equation y(x) = K(x, �)y(�) de, where B�(l - x) + Ax - A Bx(I - �)+Ax when � < x, when � > x. [Notice that the kernel obtained in this way is nonsymmetric, and discontinuous at e = x, un ess A ·- 0. 9. Transform the problem d"y dxi + xy = l, y(0) - y(l) - 0 o em egra equa 10n y(x) =f G(x, e)ey(�)de ½x(l - x), where G(x, �) = x(I - �) when x < t and G(x, ¢) = ¢(1 - x) when x > �- 11. Trans orm the problem d2y 7 +y = x, to a Fredholm integral equation. y(O) = 0, y'(l) = 0 15. Transform the problem to the relation where 1 = ¼�2 (1 - x)2 (2xe + e - 3x) when X > �- 40. (a) Show that the characteristic values of l for the equation 2r sin (x + E) E) dE are l1 = 1/.,,. and As = -1/-rr, with corresponding characteristic functions of the form y1(x) - sin x + cos x and y1(x) = sin x - cos x. (b) Obtain the most general solution of the equation Ir y(x) = l sin (x + �) y(e) d� + F(x) when F(x) = x and when F(x) = l, under the assumption that l =I= ±l/,r. (c) Prove that the equation y(x) .!_ r • sin (x + E) y(E) dE + F(x) ff 0 possesses no solution when F(x) = x, but that it possesses infinitely many solutions when F(x} = 1. Determine all such so ut1ons. (d) Determine the most general form of the prescribed function F(x), for which the integral equation f • sin (x + e) y( 0 d� = F(x), of the rst kmd, possesses a so utlon. 41. Consider the equation �y(x)�= Ftx) �A r • cos (x + e) �- (a) Determine the characteristic values of A and the corresponding char• acteristic functions. (b) Express the solution in the form ------ --ll,111-,I--1-=-=-F(x) + �•• (x,�; A)F(#ae when A is not characteristic. c) Obtain the eneral solution (when it exists) if F(x) = sin x, considerin all possible cases.