Books
in black and white
Main menu
Share a book About us Home
Books
Biology Business Chemistry Computers Culture Economics Fiction Games Guide History Management Mathematical Medicine Mental Fitnes Physics Psychology Scince Sport Technics
Ads

Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

Boyce W.E. Elementary Differential Equations and Boundary Value Problems - John Wiley & Sons, 2001. - 1310 p.
Download (direct link): elementarydifferentialequations2001.pdf
Previous << 1 .. 349 350 351 352 353 354 < 355 > 356 357 358 359 360 361 .. 609 >> Next

28. (a) Show by the method of variation of parameters that the general solution of the differential equation
/ = f (x)
can be written in the form
f x
y = (x) = c1 + c2 x I (x s) f (s) ds,
1 2 0 where c1 and c2 are arbitrary constants.
(b) Let y = (^) also be required to satisfy the boundary conditions y(0) = 0, y(1) = 0. Show that in this case
c1 = 0, c2 = 1 (1 s) f(s) ds.
(c) Show that, under the conditions of parts (a) and (b), (x) can be written in the form
(x) = ( s(1 x) f (s) ds + ( x(1 s) f (s) ds.
0x
(d) Defining
G(xs)= js(1 x), 0 - s - x,
G(x s) |x(1 s), x - s - 1,
show that the solution takes the form
(x) = ( G(x, s) f (s) ds.
0
The function G(x, s) appearing under the integral sign is a Greens function. The usefulness of a Greens function solution rests on the fact that the Greens function is independent of the nonhomogeneous term in the differential equation. Thus, once the Greens 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 (x) as given by the Greens 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
(' + ) = f (x), y(0) = 0, y(1) = 0
y = (x) = f G(x, s) f (s) ds,
0
11.3 Nonhomogeneous Boundary Value Problems
655
where
sin s sin(1 - x)
G(x, s) =
sin x
sin1
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 Greens function solution
y = () = ( G(x, s) f (s) ds, (iii)
0
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 Greens function obtained in Problem 28 is given by formula (iv).
(b) Verify that the Greens 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 = (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 Greens 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 = / (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 = (x) is given by Eq. (13), where c is defined by
Eq. (9), provided that is not an eigenvalue of the corresponding homogeneous problem.
In this case it can also be shown that the solution is given by a Greens function integral of the form
Previous << 1 .. 349 350 351 352 353 354 < 355 > 356 357 358 359 360 361 .. 609 >> Next