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
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
7 +y = x,
to a Fredholm integral equation.
y(O) = 0, y'(l) = 0 15. Transform the problem
to the relation
(1 - x)2
(2xe + e - 3x) when X > �-
40. (a) Show that the characteristic values of l for the equation
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
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)
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•
(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.
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