12
4
2
K4
K2
0
2
4
K2 Chapter K4
Figure 12S: Phase plane diagram in Problem 6, Review Section
4
y
2
K2
K1
0
1
2
x
3
2
x
K2
Figure 12T: Potential and Phase plane diagrams in Problem 8, Review Section
4 2
K4
K2
0
2
4
x
K2 K4
Figure 12U: Phase plane diagram in Problem 14, Review Section
316
CHAPTER 13: Existence and Uniqueness Theory
EXERCISES 13.1:
x
Introduction: Successive Approximations
2. y (x) =
sin[t + y (t)] dt
x
4. y (x) = 1 +
0
ey(t) dt
6. 0.3775396 8. 2.2360680 10. 1.9345632 12. y1 (x) = 1 + x; y2 (x) = 1 + x + x2 + 14. y1 (x) = y2 (x) = sin x 16. y1 (x) = y2 (x) = 3 2 5 3 x+ 2 3 1 2 x2 ; 1 6 x3 1 3 x3
x + x2
EXERCISES 13.2: 2. No 4. Yes 6. Yes
Picards Existence and Uniqueness Theorem
14. No. Let yn (x) = Then
n
n2 x ,
0 x (1/n)
2n n2 x , (1/n) x (2/n) 0 , 2/n x 1 .
lim yn (x) = 0 , 317
Chapter 13
but
1 n 0
lim
yn (x) dx = 1 = 0 .
EXERCISES 13.3: 2. [2, 1) 4. (0, 3] 6. (, ) EXERCISES 13.4: 2. 102 e 4. 102 e 6. 102 e 8. 10. e 6 1 24 esin 1
2 e1/2
Existence of Solutions of Linear Equations
Continuous Dependence of Solutions
REVIEW PROBLEMS 2. 0.7390851
x
4. 9 +
0
[t2 y 3 (t) y 2 (t)] dt
6. y1 (x) = 1 + 2x; y2 (x) = 1 + 2x 2x2 8. No 10. 12. e 6 , 22
318