—————————————————————————— CHAPTER 1. —— Chapter One Section 1.1 1. For C  "Þ&,the slopes are negative, and hence the solutions decrease. For C  "Þ&, the slopes are positive, and hence the solutions increase. The equilibrium solution appears to be C > œ "Þ&, to which all other solutions converge. a b 3. For C  "Þ&,the slopes are :9=3tive, and hence the solutions increase. For C  "Þ& , theslopes are negative, and hence the solutions decrease. All solutions appear to diverge away from the equilibrium solution C > œ "Þ&. a b 5. For C  "Î#,the slopes are :9=3tive, and hence the solutions increase. For C  "Î#, theslopes are negative, and hence the solutions decrease. All solutions diverge away from ________________________________________________________________________ page 1 —————————————————————————— CHAPTER 1. —— the equilibrium solution C > œ "Î#. a b 6. For C  #,the slopes are :9=3tive, and hence the solutions increase. For C  #, theslopes are negative, and hence the solutions decrease. All solutions diverge away from the equilibrium solution C > œ #. a b 8. For all solutions to approach the equilibrium solution C > œ #Î$, we must have a b Cw  ! for C  #Î$, and Cw  ! for C  #Î$. The required rates are satisfied by the differential equation Cw œ #$C. 9. For solutions other than C > œ # to diverge from C œ #, C > must be an increasing a b a b function for C  #, and a decreasing function for C  #. The simplest differential equation whose solutions satisfy these criteria is Cw œ C#. 10. For solutions other than C > œ "Î$ to diverge from C œ "Î$, we must have Cw  ! a b for C  "Î$, and Cw  ! for C  "Î$. The required rates are satisfied by the differential equation Cw œ $C ". 12. Note that Cw œ ! for C œ ! and C œ &. The two equilibrium solutions are C > œ ! and a b C > œ &. Based on the direction field, Cw  ! for C  &; thus solutions with initial a b values greater than & diverge from the solution C > œ &. For !  C  &, the slopes are a b negative, and hence solutions with initial values between ! and & all decrease toward the ________________________________________________________________________ page 2 —————————————————————————— CHAPTER 1. —— solution C > œ !. For C  !, the slopes are all positive; thus solutions with initial a b values less than ! approach the solution C > œ !. a b 14. Observe that Cw œ ! for C œ ! and C œ #. The two equilibrium solutions are C > œ ! a b and C > œ #. Based on the direction field, Cw  ! for C  #; thus solutions with initial a b values greater than # diverge from C > œ #. For !  C  #, the slopes are also a b positive, and hence solutions with initial values between ! and # all increase toward the solution C > œ #. For C  !, the slopes are all negative; thus solutions with initial a b values less than ! diverge from the solution C > œ !. a b 16. + Let Q > be the total amount of the drug in milligrams in the patient's body at a b a b a b any given time > 2<= . The drug is administered into the body at a constant rate of &!! a b 71Î2<Þ The rate at which the drug leaves the bloodstream is given by !Þ%Q > Þ Hence the a b accumulation rate of the drug is described by the differential equation .Q œ &!!!Þ%Q 71Î2< Þ .> a b , a b Based on the direction field, the amount of drug in the bloodstream approaches the equilibrium level of "#&!71 A3>238+0/A29?<= Þ a b 18. + Following the discussion in the text, the differential equation is a b ________________________________________________________________________ page 3 —————————————————————————— CHAPTER 1. —— .@ 7 œ 71#@# .> or equivalently, .@ # œ 1 @#Þ .> 7 , After a long time, .@ ¸ !Þ Hence the object attains a terminal velocity given by a b .> 71 @ œ Þ _ Ê # - Using the relation #@# œ 71, the required drag coefficient is # œ !Þ!%!)51Î=/-Þ a b _ . a b 19. All solutions appear to approach a linear asymptote A3>2=69://;?+6>9" . It is easy to a b verify that C > œ >$ is a solution. a b 20. ________________________________________________________________________ page 4 —————————————————————————— CHAPTER 1. —— All solutions approach the equilibrium solution C > œ !Þ a b 23. All solutions appear to diverge from the sinusoid C > œ  $ =38Ð> 1Ñ", a b # % È which is also a solution corresponding to the initial value C ! œ &Î#. a b 25. All solutions appear to converge to C > œ !. First, the rate of change is small. The a b slopes eventually increase very rapidly in magnitude. 26. ________________________________________________________________________ page 5 —————————————————————————— CHAPTER 1. —— The direction field is rather complicated. Nevertheless, the collection of points at which the slope field is zero, is given by the implicit equation C$ 'C œ #>#Þ The graph of these points is shown below: The y-intercepts of these curves are at C œ !, „ ' . It follows that for solutions with È initial values C  ' , all solutions increase without bound. For solutions with initial È values in the range C   ' and !  C  ' , the slopes remain negative, and È È hence these solutions decrease without bound. Solutions with initial conditions in the range  '  C  ! initially increase. Once the solutions reach the critical value, given by È the equation C$ 'C œ #>#, the slopes become negative and remain negative. These solutions eventually decrease without bound. ________________________________________________________________________ page 6 —————————————————————————— CHAPTER 1. —— Section 1.2 1 + The differential equation can be rewritten as a b .C œ .>Þ &C Integrating both sides of this equation results in 68 &C œ >- , or equivalently, " k k &C œ -/>. Applying the initial condition C ! œ C results in the specification of a b ! the constant as - œ &C . Hence the solution is C > œ & C & />Þ ! a b a ! b All solutions appear to converge to the equilibrium solution C > œ &Þ a b 1 - Þ Rewrite the differential equation as a b .C œ .>Þ "!#C Integrating both sides of this equation results in  "68 "!#C œ >- , or # k k " equivalently, &C œ -/#>. Applying the initial condition C ! œ C results in the specification of a b ! the constant as - œ &C . Hence the solution is C > œ & C & /#>Þ ! a b a ! b All solutions appear to converge to the equilibrium solution C > œ &, but at a faster rate a b than in Problem 1a Þ 2 + Þ The differential equation can be rewritten as a b ________________________________________________________________________ page 7 —————————————————————————— CHAPTER 1. —— .C œ .>Þ C& Integrating both sides of this equation results in 68 C& œ >- , or equivalently, " k k C& œ -/>. Applying the initial condition C ! œ C results in the specification of a b ! the constant as - œ C &. Hence the solution is C > œ & C & />Þ ! a b a ! b All solutions appear to diverge from the equilibrium solution C > œ &. a b 2 , Þ Rewrite the differential equation as a b .C œ .>Þ #C& Integrating both sides of this equation results in "68 #C& œ >- , or equivalently, # k k " #C& œ -/#>. Applying the initial condition C ! œ C results in the specification of a b ! the constant as - œ #C &. Hence the solution is C > œ #Þ& C #Þ& /#>Þ ! a b a ! b All solutions appear to diverge from the equilibrium solution C > œ #Þ&. a b 2 - . The differential equation can be rewritten as a b .C œ .>Þ #C"! Integrating both sides of this equation results in "68 #C"! œ >- , or equivalently, # k k " C& œ -/#>. Applying the initial condition C ! œ C results in the specification of a b ! the constant as - œ C &. Hence the solution is C > œ & C & /#>Þ ! a b a ! b ________________________________________________________________________ page 8 —————————————————————————— CHAPTER 1. —— All solutions appear to diverge from the equilibrium solution C > œ &. a b 3 + . Rewrite the differential equation as a b .C œ .>, ,+C which is valid for C Á ,Î+. Integrating both sides results in "68 ,+C œ >- , or + k k " equivalently, ,+C œ -/+>. Hence the general solution is C > œ ,-/+> Î+Þ a b a b Note that if C œ ,Î+, then .CÎ.> œ !, and C > œ ,Î+ is an equilibrium solution. a b , a b 3 As + increases, the equilibrium solution gets closer to C > œ !, from above. a b a b Furthermore, the convergence rate of all solutions, that is, +, also increases. 33 As , increases, then the equilibrium solution C > œ ,Î+ also becomes larger. In a b a b this case, the convergence rate remains the same. 333 If + and , both increase but ,Î+ œ constant , then the equilibrium solution a b a b C > œ ,Î+ remains the same, but the convergence rate of all solutions increases. a b 5 + . Consider the simpler equation .C Î.> œ +C . As in the previous solutions, re- a b " " write the equation as .C " œ +.>Þ C " Integrating both sides results in C > œ -/+>Þ "a b , Þ Now set C > œ C > 5, and substitute into the original differential equation. We a b a b "a b find that ________________________________________________________________________ page 9 —————————————————————————— CHAPTER 1. —— +C ! œ + C 5 ,. " a " b That is, +5 , œ !, and hence 5 œ ,Î+. - . The general solution of the differential equation is C > œ -/+> ,Î+Þ This is a b a b exactly the form given by Eq. "( in the text. Invoking an initial condition C ! œ C , a b a b ! the solution may also be expressed as C > œ ,Î+ C ,Î+ /+>Þ a b a ! b 6 + . The general solution is : > œ *!!-/>Î#, that is, : > œ *!! : *!! />Î#. a b a b a b a ! b With : œ )&!, the specific solution becomes : > œ *!!&!/>Î#. This solution is a ! a b decreasing exponential, and hence the time of extinction is equal to the number of months it takes, say > , for the population to reach zero. Solving *!!&!/>0Î# œ !, we find that 0 > œ #68 *!!Î&! œ &Þ() months. 0 a b , The solution, : > œ *!! : *!! />Î#, is a decreasing exponential as long as a b a b a ! b :  *!!. Hence *!! : *!! />0Î# œ ! has only one root, given by ! a ! b *!! > œ #68 Þ 0 Œ*!!:  ! - . The answer in part , is a general equation relating time of extinction to the value a b a b of the initial population. Setting > œ "# months, the equation may be written as 0 *!! œ /', *!!: ! which has solution : œ )*(Þ('*". Since : is the initial population, the appropriate ! ! answer is : œ )*) mice. ! 7 + . The general solution is : > œ : /<>. Based on the discussion in the text, time > is a b a b ! measured in months. Assuming " month œ $! days, the hypothesis can be expressed as : /<†" œ #: . Solving for the rate constant, < œ 68 # , with units of per month. ! ! a b , . R days œ RÎ$! months. The hypothesis is stated mathematically as : / in which 1 is the gravitational constant measured in appropriate units. The equation can be ________________________________________________________________________ page 10

Elementary Diff Eqns and Boundary-Value Problems

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