Page 62 - Advanced engineering mathematics

42 CHAPTER 1 First-Order Differential Equations

SECTION 1.6 PROBLEMS

In each of Problems 1 through 4, use Theorem 1.2 to This is a recursive definition, giving y 1 (x) in terms of y 0 ,
show that the initial value problem has a unique solution then y 2 (x) in terms of y 1 (x), and so on. The functions
in some interval about the value x 0 at which the initial con- y n (x) are called Picard iterates for the initial value prob-
dition is specified. Assume routine facts about continuity lem. Under the assumptions of the theorem, the sequence
of standard functions of two variables. of functions y n (x) converges for all x in some interval
about x 0 , and the limit of this sequence is the solution of
1. y = sin(xy); y(π/2) = 1 the initial value problem on this interval.

2. y = ln|x − y|; y(3) = π In each of Problems 6 through 9:

2
2
3. y = x − y + 8x/y; y(3) =−1

(a) Use Theorem 1.2 to show that the problem has a
xy
4. y = cos(e ); y(0) =−4

solution in some interval about x 0 .
5. Consider the initial value problem (b) Find this solution.
(c) Compute Picard iterates y 1 (x) through y 6 (x), and from

|y |= 2y; y(x 0 ) = y 0 ,
these, guess y n (x) in general.
in which x 0 is any number. (d) Find the Taylor series of the solution from part (b)
about x 0 .
(a) Assuming that y 0 > 0, find two solutions.
(b) Explain why the conclusion of part (a) does not You should find that the iterates computed in part (c)
violate Theorem 1.2. are exactly the partial sums of the series solution of
part (d). Conclude that in these examples the Picard iter-
Theorem 1.2 can be proved using Picard iterates. Here is ates converge to an infinite series representation of the
the idea. Consider the initial value problem solution.
y = f (x, y); y(x 0 ) = y 0 . 6. y = 2 − y; y(0) = 1

For each positive integer n,define 7. y = 4 + y; y(0) = 3
2

x 8. y = 2x ; y(1) = 3
y n (x) = y 0 + f (t, y n−1 (t))dt.
9. y = cos(x); y(π) = 1
x 0

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October 14, 2010 14:9 THM/NEIL Page-42 27410_01_ch01_p01-42