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Application-oriented advent relates the topic as heavily as attainable to technology. In-depth explorations of the spinoff, the differentiation and integration of the powers of x, theorems on differentiation and antidifferentiation, the chain rule and examinations of trigonometric capabilities, logarithmic and exponential features, options of integration, polar coordinates, even more.

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4 (page 341) 1 2.   x 2 + 2 x 2 1 4.   x3 e x 6 6. sec θ – sin θ tan θ + θ sin θ + (cos θ)ln(cos θ) 8. c1 x + c2 x ln x + c3 x3 − x 2  x −1   x4  x 10.   ∫ g ( x) dx +  ∫ x −5 g ( x)dx −   ∫ x −2 g ( x)dx  10   15  6     Chapter 6 Review Problems (page 344) 2. a. Lin. indep. b. Lin. indep. c. Lin. dep. 4. a. c1e−3 x + c2 e− x + c3 e x + c4 xe x ( − 2 + 5 ) x + c e( − 2 − 5 ) x b. c1e x + c2 e c. c1e x + c2 cos x + c3 sin x + c4 x cos x + c5 x sin x d. x x 1 c1e x + c2 e − x + c3 e2 x −   e x +   + 2 2 4 3  x   x   x   x  x/ 2 x/ 2 −x / 2 2 6.

Passes through equilibrium at time 2π 5 20 5 36 CHAPTER 4 LINEAR SECOND-ORDER EQUATIONS 4. 896. 8 b = 16: y (t ) = (1 + 8t )e−8t Figure 13 (b = 16) 4 1 b = 20: y (t ) =   e −4t −   e −16t 3 3 Figure 14 (b = 20) 37 38 CHAPTER 4 LINEAR SECOND-ORDER EQUATIONS 1 + 2  ( −2 + 2 )t 1 − 2  ( −2 − 2 )t 6. 615 .  2  Figure 17 (k = 6) 8. Never 10. 755 m 12. 9 (page 227) 2. M (γ ) = 1 (3 − 2γ 2 ) 2 + 9γ 2 Figure 18 Response curve for Problem 2  5  4. y (t ) = 1 +   t  sin t  2  16.

Y = cx 2 / 3 Figure 22 47 48 CHAPTER 5 INTRODUCTION TO SYSTEMS AND PHASE PLANE ANALYSIS 16. (0, 0) is a stable node. Figure 23 18. (0, 0) is an unstable node. (0, 5) is a stable node. (7, 0) is a stable node. (3, 2) is a saddle point. 4 20. {vy′′ == −v y ; (0, 0) is a center. Figure 25  y′ = v 22.  3 ; (0, 0) is a center. v ′ = − y Figure 26 49 50 CHAPTER 5 INTRODUCTION TO SYSTEMS AND PHASE PLANE ANALYSIS  y′ = v 24.  3; v ′ = − y + y (0, 0) is a center. (–1, 0) is a saddle point. (1, 0) is a saddle point.