# 1 d d y x 5 x 4 sec 2 x y x 5 tan x c 2 d d y x sec x

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1. d d y x 5 x 4 sec 2 x y x 5 tan x C 2. d d y x sec x tan x e x y sec x e x C 3. d d y x sin x e x 8 x 3 y cos x e x 2 x 4 C 4. d d y x 1 x x 1 2 ( x 0) y ln x x 1 C 5. d d y x 5 x ln 5 x 2 1 1 y 5 x tan 1 x C 6. d d y x 1 1 x 2 1 x y sin 1 x 2 x C 7. d d y t 3 t 2 cos( t 3 ) y sin ( t 3 ) C 8. d d y t (cos t ) e sin t y e sin t C 9. d d u x (sec 2 x 5 )(5 x 4 ) u tan( x 5 ) C 10. d d u y 4(sin u ) 3 (cos u ) y (sin u ) 4 C In Exercises 11–20, solve the initial value problem explicitly. 11. d d y x 3 sin x and y 2 when x 0 y 3 cos x 5 12. d d y x 2 e x cos x and y 3 when x 0 y 2 e x sin x 1 Section 6.1 Exercises 6. d d y x 1 y y 2 x Yes 7. d d y x y tan x y sec x Yes 8. d d y x y 2 y x 1 No In Exercises 9–12, find the constant C. 9. y 3 x 2 4 x C and y 2 when x 1 5 10. y 2sin x 3 cos x C and y 4 when x 0 7 11. y e 2 x sec x C and y 5 when x 0 3 12. y tan 1 x ln(2 x 1) C and y p when x 1 3 p 4 13. d d u x 7 x 6 3 x 2 5 and u 1 when x 1 u x 7 x 3 5 x 4 14. d d A x 10 x 9 5 x 4 2 x 4 and A 6 when x 1 15. d d y x x 1 2 x 3 4 12 and y 3 when x 1 16. d d y x 5 sec 2 x 3 2 x and y 7 when x 0 17. d d y t 1 1 t 2 2 t ln 2 and y 3 when t 0 18. d d x t 1 t t 1 2 6 and x 0 when t 1 19. d d v t 4 sec t tan t e t 6 t and v 5 when t 0 20. d d s t t (3 t 2) and s 0 when t 1 In Exercises 21–24, solve the initial value problem using the Funda- mental Theorem. (Your answer will contain a definite integral.) 21. d d y x sin ( x 2 ) and y 5 when x 1 y x 1 sin( t 2 ) dt 5 22. d d u x 2 cos x and u 3 when x 0 23. F ( x ) e cos x and F (2) 9 F ( x ) x 2 e cos t dt 9 24. G ( s ) 3 tan s and G (0) 4 G ( s ) s 0 3 tan t dt 4 A x 10 x 5 x 2 4 x 1 y x 1 x 3 12 x 11 ( x 0) y 5 tan x x 3 2 7 (0 x < p 2) y tan 1 t 2 t 2 x ln t t 1 6 t 7 ( t 0) v 4 sec t e t 3 t 2 ( p 2 t p 2) (Note that C 0.) s t 3 t 2 (Note that C 0.) u x 0 2 cos t dt 3
328 Chapter 6 Differential Equations and Mathematical Modeling In Exercises 25–28, match the differential equation with the graph of a family of functions (a)–(d) that solve it. Use slope analysis, not your graphing calculator. 25. d d y x (sin x ) 2 Graph (b) 26. d d y x (sin x ) 3 Graph (c) 27. d d y x (cos x ) 2 Graph (a) 28. d d y x (cos x ) 3 Graph (d) In Exercises 29–34, construct a slope field for the differential equation. In each case, copy the graph at the right and draw tiny segments through the twelve lattice points shown in the graph. Use slope analysis, not your graphing calculator. 29. d d y x x 30. d d y x y 31. d d y x 2 x y 32. d d y x 2 x y 33. d d y x x 2 y 34. d d y x x 2 y In Exercises 35–40, match the differential equation with the appropri- ate slope field. Then use the slope field to sketch the graph of the par- ticular solution through the highlighted point (3, 2). (All slope fields are shown in the window [ 6, 6] by [ 4, 4].) (d) (c) (b) (a) (d) (c) (b) (a) 35. d d y x x 36. d d y x y 37. d d y x x y 38. d d y x y x 39. d d y x y x 40. d d y x x y In Exercises 41–44, use Euler’s Method with increments of Δ x 0.1 to approximate the value of y when x 1.3. 41. d d y x x 1 and y 2 when x 1 2.03 42. d d y x y 1 and y 3 when x 1 3.662 43.