# Elementary differential equations 7th edition - Boyce W.E

ISBN 0-471-31999-6

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y(0) = 0, /(1) = 0; see Section 11.2, Problem 7.

/(0) = 0, y (1) = 0; see Section 11.2, Problem 3.

/(0) = 0, y (1) + y(1) = 0; see Section 11.2, Problem 11.

- 2x|, y(0) = 0, y(1) = 0

In each of Problems 6 through 9 determine a formal eigenfunction series expansion for the solution of the given problem. Assume that f satisfies the conditions of Theorem 11.3.1. State the values of p for which the solution exists.

6. y + py =-f(x), y(0) = °, Z(1) = 0

7. y + py =-f(x), y(0) = 0, y(1) = 0

8. y + py =-f(x), y(0) = 0, y(1) = 0

9. y + py =-f(x), y(0) = 0, y(1) + y(1) = 0

In each of Problems 10 through 13 determine whether there is any value of the constant a for which the problem has a solution. Find the solution for each such value.

10. y + n2y = a + x, y(0) = 0, y(1) = 0

11. y + 4n2y = a + x, y(0) = 0, y(1) = 0

12. y + n2y = a, y(0) = 0, /(1) = 0

13. y + n2y = a — cos nx, y(0) = 0, y(1) = 0

14. Let 01,..., , ... be the normalized eigenfunctions of the differential equation (3) subject

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to the boundary conditions (2). If ^ cn(x) converges to f (x), where f (x) = 0 for each

n=1

x in 0 < x < 1, show that cn = 0 for each n.

Hint: Multiply by r(x)0m(x), integrate, and use the orthogonality property of the eigenfunctions.

15. Let L be a second order linear differential operator. Show that the solution y = 0(x) of the problem

L [y] = f (x), a1 y(0) + a2 y (0) = a, b1 y(1) + b2 y (1) = p

can be written as y = u + v, where u = 01 (x) and v = 02 (x) are solutions of the problems

L [u] = 0,

a1 u(0) + a2u'(0) = a, b1 u(1) + b2u'(1) = p

and

L [v] = f (x), a1v(0) + a2v'(0) = 0, b1v(1) + b2v'(1) = 0,

respectively.

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Chapter 11. Boundary Value Problems and Sturm Liouville Theory

16. Show that the problem

y" + n2 y = n2 x, y(0) = 1, y(1) = 0

has the solution

y = c1 sin n x + cos n x + x.

Also show that this solution cannot be obtained by splitting the problem as suggested in Problem 15, since neither of the two subsidiary problems can be solved in this case.

17. Consider the problem

y + p(x)y + q (x)y = 0, y(0) = a, y(1) = b.

Let y = u + v, where v is any twice differentiable function satisfying the boundary conditions (but not necessarily the differential equation). Show that u is a solution of the problem

u" + p(x)d + q (x)u = g(x), u(0) = 0, u(1) = 0,

where g(x) = —[v" + p(x)v' + q(x)v], and is known once v is chosen. Thus nonhomogeneities can be transferred from the boundary conditions to the differential equation. Find a function v for this problem.

18. Using the method of Problem 17, transform the problem

y + 2y = 2 — 4x, y(0) = 1, y(1) + y (1) = -2

into a new problem in which the boundary conditions are homogeneous. Solve the latter problem by reference to Example 1 of the text.

In each of Problems 19 through 22 use eigenfunction expansions to find the solution of the given boundary value problem.

19. ut = uxx — x, u(0, t) = 0, ux(1, t) = 0, u(x, 0) = sin(nx/2);

see Problem 2.

20. ut = uxx + e—t, ux(0, t) = 0, ux(1, t) + u(1, t) = 0, u(x, 0) = 1 — x;

see Section 11.2, Problems 10 and 12.

21. ut = uxx + 1 — |1 — 2x|, u(0, t) = 0, u(1, t) = 0, u(x, 0) = 0;

see Problem 5.

22. ut = uxx + e—t(1 — x), u(0, t) = 0, ux(1, t) = 0, u(x, 0) = 0;

see Section 11.2, Problems 6 and 7.

23. Consider the boundary value problem

r (x)ut = [ p(x)ux ]x — q (x)u + F(x), u(0, t) = Tp u(1, t) = T2, u(x, 0) = f (x).

(a) Let v(x) be a solution of the problem

[ p(x)v']' — q (x)v = —F(x), v(0) = Tj, v(1) = T2.

If w(x, t) = u(x, t) — v(x), find the boundary value problem satisfied by w. Note that this problem can be solved by the method of this section.

(b) Generalize the procedure of part (a) to the case where u satisfies the boundary conditions

ux(0, t) — h1u(0, t) = Tv ux(1, t) + h2u(1, t) = T2.

In each of Problems 24 and 25 use the method indicated in Problem 23 to solve the given boundary value problem.

11.3 Nonhomogeneous Boundary Value Problems

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24. u=u —2, 25. u=u —n2 cos nx,

t xx 9 t xx 9

u(0, t) = 1, u(1, t) = 0, ux(0, t) = 0, u(l, t) = 1,

u(x, 0) = x2 — 2x + 2 u(x, 0) = cos(3nx/2) — cos nx

26. The method of eigenfunction expansions is often useful for nonhomogeneous problems related to the wave equation or its generalizations. Consider the problem

r(x)utt = [p(x)ux]x — q(x)u + F(x, t), (i)

ux(0, t) — h1u(0, t) = 0, ux(1, t) + h2u(1, t) = 0, (ii)

u(x, 0) = f (x), ut(x, 0) = g(x). (iii)

This problem can arise in connection with generalizations of the telegraph equation (Problem 16 in Section 11.1) or the longitudinal vibrations of an elastic bar (Problem 25 in Section 11.1).

(a) Let u(x, t) = X(x)T(t) in the homogeneous equation corresponding to Eq. (i) and show that X(x) satisfies Eqs. (28) and (29) of the text. Let kn and 0n(x) denote the eigenvalues and normalized eigenfunctions of this problem.

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(b) Assume that u(x, t) = ^ b (t)$ (x), and show that b (t) must satisfy the initial value

n=1 n n n

problem

b'n(t) + *nbn (t) = Yn (t), bn (0) = aB, b'n (0) = pn,

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