Elementary Differential Equations 8th edition by Boyce ch01

Elementary Differential Equations, with ODE Architect CD

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—————————————————————————— —— CHAPTER 1. ________________________________________________________________________ page 1 Chapter One Section 1.1 1. For , the slopes are C  "Þ& negative , and hence the solutions decrease. For , the C  "Þ& slopes are , and hence the solutions increase. The equilibrium solution appears to positive be , to which all other solutions converge. C > œ "Þ& a b 3. For , the slopes are C   "Þ& :9=3 tive , and hence the solutions increase. For C   "Þ& , the slopes are , and hence the solutions decrease. All solutions appear to negative diverge away from the equilibrium solution . C > œ  "Þ& a b 5. For , the slopes are C   "Î# :9=3 tive , and hence the solutions increase. For C   "Î# , the slopes are , and hence the solutions decrease. All solutions negative diverge away from
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—————————————————————————— —— CHAPTER 1. ________________________________________________________________________ page 2 the equilibrium solution . C > œ  "Î# a b 6. For , the slopes are C   # :9=3 tive , and hence the solutions increase. For , C   # the slopes are , and hence the solutions decrease. All solutions diverge away negative from the equilibrium solution . C > œ  # a b 8. For solutions to approach the equilibrium solution , we must have all C > œ #Î$ a b C  ! C  #Î$ C  ! C  #Î$ w w for , and for . The required rates are satisfied by the differential equation . C œ #  $C w 9. For solutions than to diverge from , must be an other increasing C > œ # C œ # C > a b a b function for , and a function for . The simplest differential C  # C  # decreasing equation whose solutions satisfy these criteria is . C œ C  # w 10. For solutions than to diverge from , we must have other C > œ "Î$ C œ "Î$ C  ! a b w for , and for . The required rates are satisfied by the differential C  "Î$ C  ! C  "Î$ w equation . C œ $C  " w 12. Note that for and . The two equilibrium solutions are and C œ ! C œ ! C œ & C > œ ! w a b C > œ & C  ! C  & a b . Based on the direction field, for ; thus solutions with initial w values than diverge from the solution . For , the slopes are greater & C > œ & !  C  & a b negative between , and hence solutions with initial values and all decrease toward the ! &
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—————————————————————————— —— CHAPTER 1. ________________________________________________________________________ page 3 solution . For , the slopes are all ; thus solutions with initial C > œ ! C  ! a b positive values less than approach the solution . ! C > œ ! a b 14. Observe that for and . The two equilibrium solutions are C œ ! C œ ! C œ # C > œ ! w a b and . Based on the direction field, for ; thus solutions with initial C > œ # C  ! C  # a b w values than diverge from . For , the slopes are also greater # C > œ # !  C  # a b positive between , and hence solutions with initial values and all increase toward the ! # solution C > œ # C  ! a b . For , the slopes are all ; thus solutions with initial negative values than diverge from the solution . less ! C > œ ! a b 16. Let be the total amount of the drug in the patient's body at a b a b a b + Q > in milligrams any given time . The drug is administered into the body at a rate of > 2<= &!! a b constant 71Î2<Þ
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