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

ISBN 0-471-31999-6

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? 18. Consider the initial value problem

where a is a given number.

(a) Draw a direction field for the differential equation (or reexamine the one from Problem 8). Observe that there is a critical value of a in the interval 2 < a < 3 that separates converging solutions from diverging ones. Call this critical value a0.

(b) Use Euler’s method with h = 0.01 to estimate a0. Do this by restricting a0 to an interval [a, b], where b — a = 0.01.

? 19. Consider the initial value problem

where a is a given number.

(a) Draw a direction field for the differential equation. Observe that there is a critical value of a in the interval 0 < a < 1 that separates converging solutions from diverging ones. Call this critical value a0.

(b) Use Euler’s method with h = 0.01 to estimate a0. Do this by restricting a0 to an interval [a, b], where b — a = 0.01.

20. Convergence of Euler’s Method. It can be shown that, under suitable conditions on f, the numerical approximation generated by the Euler method for the initial value problem ? = f (t, y), y(t0) = y0 converges to the exact solution as the step size h decreases. This is illustrated by the following example. Consider the initial value problem

y = t2 + y2, y(0) = 1.

y = (y2 + 2ty)/(3 + t2), y (1) = 2.

? = ~ty + 0.1 y3, y(0) = a,

/ = y2 - t2, y(0) = a,

y= 1 - f + y, y(to) = yo.

(a) Show that the exact solution is y = ?(t) = (y0 — t0)el ^ + t.

(b) Using the Euler formula, show that

yk = (1 + h) y— + h — htk—1, k = 1 2,....

(c) Noting that y1 = (1 + h)(y0 — t0) + t1, show by induction that

yn = (1 + h)n (y0 — t0) + tn

(1)

for each positive integer n.

2.8 The Existence and Uniqueness Theorem

105

(d) Consider a fixed point t > to and for a given n choose h = (t — t0)/n. Then tn = t for every n. Note also that h ^ 0 as n ^?. By substituting for h in Eq. (i) and letting n ^ ?, show that yn ^ ?(t) as n ^?.

Hint: lim (1 + a/n)n = ea.

n^?

In each of Problems 21 through 23 use the technique discussed in Problem 20 to show that the approximation obtained by the Euler method converges to the exact solution at any fixed point as h ^ 0.

21. y = y, y(0) = 1

22. y/ = 2y — 1, y(0) = 1 Hint: y1 = (1 + 2h)/2 + 1/2

23. / = 2 — t + 2y, y(0) = 1 Hint: y1 = (1 + 2h) + t1/2

2.8 The Existence and Uniqueness Theorem

In this section we discuss the proof of Theorem 2.4.2, the fundamental existence and uniqueness theorem for first order initial value problems. This theorem states that under certain conditions on f ( t, y), the initial value problem

y= f (t, y), y(t0) = y0 (1)

has a unique solution in some interval containing the point t0.

In some cases (for example, if the differential equation is linear) the existence of a solution of the initial value problem (1) can be established directly by actually solving the problem and exhibiting a formula for the solution. However, in general, this approach is not feasible because there is no method of solving the differential equation that applies in all cases. Therefore, for the general case it is necessary to adopt an indirect approach that demonstrates the existence of a solution of Eqs. (1), but usually does not provide a practical means of finding it. The heart of this method is the construction of a sequence of functions that converges to a limit function satisfying the initial value problem, although the members of the sequence individually do not. As a rule, it is impossible to compute explicitly more than a few members of the sequence; therefore, the limit function can be determined only in rare cases. Nevertheless, under the restrictions on f (t, y) stated in Theorem 2.4.2, it is possible to show that the sequence in question converges and that the limit function has the desired properties. The argument is fairly intricate and depends, in part, on techniques and results that are usually encountered for the first time in a course on advanced calculus. Consequently, we do not go into all of the details of the proof here; we do, however, indicate its main features and point out some of the difficulties involved.

First of all, we note that it is sufficient to consider the problem in which the initial point (t0, y0) is the origin; that is, the problem

/= f(t, y), y(0) = 0. (2)

If some other initial point is given, then we can always make a preliminary change of variables, corresponding to a translation of the coordinate axes, that will take the given point (t0, y0) into the origin. The existence and uniqueness theorem can now be stated in the following way.

106

Chapter 2. First Order Differential Equations

Theorem 2.8.1

If f and d f/d y are continuous in a rectangle R: \t \ < a, \ y\< b, then there is some interval \t\< h < a in which there exists a unique solution y = 0(t) of the initial value problem (2).

To prove this theorem it is necessary to transform the initial value problem (2) into a more convenient form. If we suppose temporarily that there is a function y = 0(t) that satisfies the initial value problem, then f [t, 0(t)] is a continuous function of t only. Hence we can integrate / = f (t, y) from the initial point t = 0 to an arbitrary value of t, obtaining

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