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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
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(2cosVX~ — 1) cos JX~ x .TO sin(n—/2) sin n— x
4. y = 2> ---------------12 ___________________ 5. y = 8 σ ¦
=0 Xn (Xn - 2)(1 + sin2 ) n=0 (n2—2 - 2)n2—2
6-9. For each problem the solution is
“ c f1
y = J2 τn(x), cn = f (x)τ- (x) dx, j = x-,
n=1 Xn - J J0
where Το(x) is given in Problems 1, 2, 3, and 4, respectively, in Section 11.2, and Xn is the corresponding eigenvalue. In Problem 8 summation starts at n = 0.
1 1 1 (
10. a = —, y=—:r cos — x^-----------------------------------2 x-+csin—x
2 2—2 —2 2
11. No solution 12. a is arbitrary, y = ccos — x + a/—2
13. a = 0, y = csin — x — (x/2—) sin—x 17. v(x) = a + (b — a)x
2 x
18. v(x) = 1 - 2x
19. u(x, t) = V2
- 44+(—o+-°ji e-—2»4
——
—x
sin------
2
TO 4c 0 0
- -Μn 0)2—2 _ e-(n-1/2) — f] sin(n - 2)—x,
n=2 (2n - 0) —
4-2(—1)n+1
n = 1, 2,...
n (2n - 1)2—
TO Γ c
20. u(x, t) = -2^] '
n=1
.Xn - 1
(e-f - e-V) + ΰε-
(1 + sin2 )1/2’
42 sin 42(1 - cos^X)
n (1 + sin2 συ-)1/2’ n x-(1 + sm2yx-)1/2’
and Xn satisfies cos — yX sin yT- = 0.
sin(n— /2) _n2 — 2 f
21. u(x, t) = 8 / -----— (1 — e n f) sinn— x
-To n —
c (e- f - e- (n—1/2)2—2f) sin(n - 1)— x
22. u(x, t) = 72^ ^^-,
-=0 (- - 0 )2—2 -1
2—2(2- - 1)— + 4—2(-1)n
c = ----------------------------------
n (2n - 1)2—2
23. (a)r(x)wt = [p(x)wx]x — q(x)w, w(0, t) = 0, w(1, t) = 0, w(x, 0) = f(x) — v(x)
2 4 -TO e—(2n—1)2—2f sin(2n - 1)—x
24. u(x, t) = x2 - 2x + 1 + -> ----------------
— 2n - 1
n=1
25. u(x, t) = — cos — x + e-9— t/4cos(3—x/2)
31-34. In all cases solution is y = I G(x, s) f (s) ds, where G(x, s) is given below.
0
31. G(x, s) = j° - x, 0 < s < x 32. G(x, s) = js(2 - ^ 0 < s < x
[1 — s, x < s < 1 lx(2 — s)/2, x < s < 1
cos
734
Answers to Problems
See SSM for detailed solutions to 33
2abc
4ab
1abc
2
4, 5ab, 6 7b, 9a, 10
cos s sin(1 x)/ cos 1, 0 s x
33. G(x, s) = { . \ , ’ _ “
[sm(1 — s) cos x/ cos1, x < s < 1
34. G(x, s) = js, 0 < s < x
x, x s 1
Section 11.4, page 661
c p 1 j p 1
1. y =J2J0(VKxϊ c- = f(x)MVKx) dx xJo(^fKx) dx,
n 1 Xn δ 0 0
Xn δ
JTn satisfies J0 (\/~X) = 0 c TO c 2 (c) σ = — ³ + E ^ MVKx);
' n=1 n ^
c0 = 2 / f (x) dx; cn = ^ f (x) J0(.^Xnx) dx j xj0!(^^x) dx, n = 1, 2,...;
^/X"- satisfies J0 (VX) = 0
3. (d) an = / xJ (T>) f (x) dx j ^ xJ (jX~nx) dx
TO c 1 I „ 1
(e)y= DJ(^x), c- = f(x)J(VKx) dx xJJ^^Vr-x) dx
n 1 Xn δ 0 0
4. (b) y = f] P2n — 1(x), cn =/ f (x) P2n — 1(x) d^ / P2n— 1(x) dx
n=1 Xn δ 0 0
n=1 Xn δ
Section 11.5, page 666
1. (b) u(?, 2) = f(? + 1), u(?, 0) = 0, 0 < ? < 2
u(0, n) = u(2, n) = 0, 0 < n < 2
TO 1 /*1 / /*1
2. 0 = J] knJ0 (Xnr) sin kn = 1------ rJ0(Xnr)g(r) dr/ rJ0(Xnr) dr
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