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

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Elementary Differential Equations 8th edition by Boyce ch01...

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