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28. (a) Show by the method of variation of parameters that the general solution of the differential equation
ó/ = f (x)
can be written in the form
y = Ű(x) = c1 + c2 x óI (x ó s) f (s) ds,
1 2 0 where c1 and c2 are arbitrary constants.
(b) Let y = Ű(^) also be required to satisfy the boundary conditions y(0) = 0, y(1) = 0. Show that in this case
c1 = 0, c2 = 1 (1 ó s) f(s) ds.
(c) Show that, under the conditions of parts (a) and (b), Ű (x) can be written in the form
Ű(x) = ( s(1 ó x) f (s) ds + ( x(1 ó s) f (s) ds.
G(xs)= js(1 ó x), 0 - s - x,
G(xí s) |x(1 ó s), x - s - 1,
show that the solution takes the form
Ű (x) = ( G(x, s) f (s) ds.
The function G(x, s) appearing under the integral sign is a Greenís function. The usefulness of a Greenís function solution rests on the fact that the Greenís function is independent of the nonhomogeneous term in the differential equation. Thus, once the Greenís function is determined, the solution of the boundary value problem for any nonhomogeneous term f (x) is obtained by a single integration. Note further that no determination of arbitrary constants is required, since Ű (x) as given by the Greenís function integral formula automatically satisfies the boundary conditions.
29. By a procedure similar to that in Problem 28 show that the solution of the boundary value problem
ó (”' + ”) = f (x), y(0) = 0, y(1) = 0
y = Ű (x) = f G(x, s) f (s) ds,
11.3 Nonhomogeneous Boundary Value Problems
sin s sin(1 - x)
G(x, s) =
30. It is possible to show that the Sturm-Liouville problem
sin1 sin x sin(1 - s)
0 < s < x, x < s < 1.
L [y] = -[ p(x)/]' + q (x) y = f (x), (i)
a1 y(0) + a2 y (0) = 0, b1 y(1) + b2/(1) = 0 (ii)
has a Greenís function solution
y = Ű(Ž) = ( G(x, s) f (s) ds, (iii)
provided that X = 0 is not an eigenvalue of L[y] = Xy subject to the boundary conditions
(ii). Further, G(x, s) is given by
\-”J(s)”2(x)/p(x) W(”1, y2)(x), 0 < s < x,
G(x, s) = (iv)
(-y1 (x)y2(s)/p(x) W(y1, y2)(x), x < s < 1,
where y1 is a solution of L[ y] = 0 satisfying the boundary condition at x = 0, y2 is a solution of L[y] = 0 satisfying the boundary condition at x = 1, and W(y1, y2) is the Wronskian of y and y2.
(a) Verify that the Greenís function obtained in Problem 28 is given by formula (iv).
(b) Verify that the Greenís function obtained in Problem 29 is given by formula (iv).
(c) Show that p(x) W(y1, y2)(x) is a constant, by showing that its derivative is zero.
(d) Using Eq. (iv) and the result of part (c), show that G(x, s) = G(s, x).
(e) Verify that y = Ű (x) from Eq. (iii) with G(x, s) given by Eq. (iv) satisfies the differential equation (i) and the boundary conditions (ii).
In each of Problems 31 through 34 solve the given boundary value problem by determining the appropriate Greenís function and expressing the solution as a definite integral. Use Eqs. (i) to (iv) of Problem 30.
31. - ”'= f(x), y (0) = 0, y(1) = 0
32. - ”' = f (x), y(0) = 0, y(l) + y>(1) = 0
33. -(” + y) = f(x), y(0) = 0, y(1) = 0
34. - ” = f (x), y(0) = 0, y(1) = 0
35. Consider the boundary value problem
L [y] = -[ p(x)y] + q (x) y = /Ž„ (x) y + f (x), (i)
a1 y(0) + a2 y (0) = 0, b1 y(1) + b2/0) = 0. (ii)
According to the text, the solution y = Ű (x) is given by Eq. (13), where cĄ is defined by
Eq. (9), provided that Ž is not an eigenvalue of the corresponding homogeneous problem.
In this case it can also be shown that the solution is given by a Greenís function integral of the form