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

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> 11. y = — sin y + 1, y(0) = 0

> 12. y = (3t2 + 4t + 2)/2(y — 1), y(0) = 0

13. Let ôï (x) = õï for 0 < x < 1 and show that

r a , \ (0, 0 < x < 1,

lim ôï(x) = 1 ,

ï^æ 11 I 1, x = 1 .

114

Chapter 2. First Order Differential Equations

This example shows that a sequence of continuous functions may converge to a limit function that is discontinuous.

_nx2

14. Consider the sequence ôï(x) = 2ïõå , 0 — x — 1.

(a) Show that lim ô (x) = 0 for 0 — x — 1; hence

ï^æ ï

f lim ô (x) dx = 0.

Jo ï^æ

(b) Show that f 2nxe—nx dx = 1 — å—ï; hence

J0

lim [ ôï (

ï^æJ 0

lim I ôï (x) dx = 1.

Thus, in this example,

n b n b

lim ² ô (x) dx = I lim ô (x) dx,

ï^æJa ï Ja ï^æ

even though lim ôï (x) exists and is continuous.

ï^æ ï

In Problems 15 through 18 we indicate how to prove that the sequence {ôï (t)}, defined by Eqs. (4) through (7), converges.

15. If ä f/äy is continuous in the rectangle D, show that there is a positive constant Ê such that

I f(t,Ó1) — f(t,Ó2)I — ÊIÓ1 — Ó2²,

where (t,y1) and (t,y2) are any two points in Dhaving the same t coordinate. This inequality is known as a Lipschitz condition.

Hint: Hold t fixed and use the mean value theorem on f as a function of y only. Choose Ê to be the maximum value of |d f/äyl in D.

16. If ô 1 (t) and ôï(t) are members of the sequence {ôï(t)}, use the result of Problem 15 to show that

I f [t, ôï (t)] — f [t, ôï_1Ø — Ê ²ôï (t) — ôï-AOl

17. (a) Show that if It l — h, then

²ô()² — M It I,

where Mis chosen so that I f(t,y)I — Mfor (t,y) in D.

(b) Use the results of Problem 16 and part (a) of Problem 17 to show that

M^t I2

ô2(t) — ôóò — -^-.

(c) Show, by mathematical induction, that

M.Êï—1\³ Iï M^h’1

rnn (t) — ôï-1(t)I — —^ — —!—.

— ï! ï!

18. Note that

ôï (t) = ô1 (t) + [ô2(0 — ô1 (t)] +-----+ [ôï (t) — ôï—1 (t)].

(a) Show that

I(Pn (t)I — \<P1(t)I + I(P2(t) — <P1(t)I + ••• + I(Pn (t) — <Pn_1(t)I.

2.9 First Order Difference Equations

115

(b) Use the results of Problem 17 to show that

M

ô (t)I — Ê

(Kh)2 (Kh)n

Kh + -—- + ••• + ( )

2! ï!

(c) Show that the sum in part (b) converges as n ^æ and, hence, the sum in part (a) also converges as n ^æ. Conclude therefore that the sequence {ôï(t)} converges since it is the sequence of partial sums of a convergent infinite series.

19. In this problem we deal with the question of uniqueness of the solution of the integral equation (3),

ô(^) = f f[s^(s)] ds.

0

(a) Suppose that ô and ty are two solutions ofEq. (3). Show that, for t > 0,

ty(t) — ty(t) = { f[s^(s)] — f[s, ty(s)]} ds.

0

(b) Show that

rn(t) — ty(t)I — / I f[s, ô()] — f[s, ty(s)]I ds.

0

(c) Use the result of Problem 15 to show that

t{ f[s,

0

f ‘ I f [s,

0

^(t) — ty(t)I — ê( ^(s) — ty(s)I ds,

0

where Ê is an upper bound for Id f/ä yI in D. This is the same as Eq. (30), and the rest of the proof may be constructed as indicated in the text.

2.9 First Order Difference Equations

©While a continuous model leading to a differential equation is reasonable and attractive for many problems, there are some cases in which a discrete model may be more natural. For instance, the continuous model of compound interest used in Section 2.3 is only an approximation to the actual discrete process. Similarly, sometimes population growth may be described more accurately by a discrete than by a continuous model. This is true, for example, of species whose generations do not overlap and that propagate at regular intervals, such as at particular times of the calendar year. Then the population yn+1 of the species in the year n + 1 is some function of n and the population yn in the preceding year, that is,

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