Separation of Variables and the Logistic Equation. Separation of Variables. M x N y dy. x 2 3y dy. sin x y cos x. dy cot x dx 1 e y 1 dy 2 x dx

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1 SECTION 63 Separation of Variables and the Logistic Equation 2 Section 63 Separation of Variables and the Logistic Equation Recognize and solve differential equations that can be solved b separation of variables Recognize and solve homogeneous differential equations Use differential equations to model and solve applied problems Solve and analze logistic differential equations Separation of Variables Consider a differential equation that can be written in the form M N d d 0 where M is a continuous function of alone and N is a continuous function of alone As ou saw in the preceding section, for this tpe of equation, all terms can be collected with d and all terms with d, and a solution can be obtained b integration Such equations are said to be separable, and the solution procedure is called separation of variables Below are some eamples of differential equations that are separable Original Differential Equation 2 3 d d 0 sin cos e 2 Rewritten with Variables Separated 3 d 2 d d cot d e d 2 d EXAMPLE Separation of Variables Find the general solution of 2 d d NOTE Be sure to check our solutions throughout this chapter In Eample, ou can check the solution C 2 b differentiating and substituting into the original equation 2 d d C 2 2? C 2 C 2 C 2 So, the solution checks Solution To begin, note that 0 is a solution To find other solutions, assume that 0 and separate variables as shown 2 d d Differential form d Separate variables 2 d Now, integrate to obtain d 2 d ln 2 ln 2 C ln ln 2 C ec 2 ±e C 2 Integrate Because 0 is also a solution, ou can write the general solution as C 2 General solution Tr It Eploration A Open Eploration The editable graph feature below allows ou to edit the graph of a function Editable Graph

2 22 CHAPTER 6 Differential Equations In some cases it is not feasible to write the general solution in the eplicit form f The net eample illustrates such a solution Implicit differentiation can be used to verif this solution FOR FURTHER INFORMATION For an eample (from engineering) of a differential equation that is separable, see the article Designing a Rose Cutter b J S Hartzler in The College Mathematics Journal MathArticle EXAMPLE 2 Finding a Particular Solution Given the initial condition 0, find the particular solution of the equation d e 2 2 d 0 Solution Note that 0 is a solution of the differential equation but this solution does not satisf the initial condition So, ou can assume that 0 To separate variables, ou must rid the first term of and the second term of e 2 So, ou should multipl b e 2 and obtain the following d e 2 2 d 0 e 2 2 d d d e 2 d From the initial condition 0, ou have C, which implies that C So, the particular solution has the implicit form 2 2 ln 2 e2 2 ln 2 e ln 2 e2 C You can check this b differentiating and rewriting to get the original equation Tr It EXAMPLE 3 Eploration A Finding a Particular Solution Curve 0 = 3e Find the equation of the curve that passes through the point, 3 and has a slope of 2 at an point, Solution Because the slope of the curve is given b 2, ou have d d 2 with the initial condition 3 Separating variables and integrating produces d d 2, Figure 6 (, 3) = 3e ( )/ ln C e C Ce Because 3 when, it follows that 3 Ce and C 3e So, the equation of the specified curve is 3e e 3e, > 0 See Figure 6 Editable Graph Tr It Eploration A

3 SECTION 63 Separation of Variables and the Logistic Equation 23 NOTE The notation f, is used to denote a function of two variables in much the same wa as f denotes a function of one variable You will stud functions of two variables in detail in Chapter 3 Homogeneous Differential Equations Some differential equations that are not separable in and can be made separable b a change of variables This is true for differential equations of the form where f is a homogeneous function The function given b f, is homogeneous of degree n if f t, t t n f, Homogeneous function of degree n where n is a real number f,, EXAMPLE Verifing Homogeneous Functions a f, is a homogeneous function of degree 3 because f t, t t 2 t t 3 3 t t 2 t 3 2 t 3 3 t t t 3 f, b f, e sin is a homogeneous function of degree because f t, t te t t t sin t t t e sin tf, c f, 2 is not a homogeneous function because f t, t t t 2 2 t t 2 t n 2 d f, is a homogeneous function of degree 0 because f t, t t t t 0 Tr It Eploration A Definition of Homogeneous Differential Equation A homogeneous differential equation is an equation of the form M, d N, d 0 where M and N are homogeneous functions of the same degree EXAMPLE 5 Testing for Homogeneous Differential Equations a 2 d 2 d 0 is homogeneous of degree 2 b 3 d 3 d is homogeneous of degree 3 c 2 d 2 d 0 is not a homogeneous differential equation Tr It Eploration A

4 2 CHAPTER 6 Differential Equations To solve a homogeneous differential equation b the method of separation of variables, use the following change of variables theorem THEOREM 62 Change of Variables for Homogeneous Equations If M, d N, d 0 is homogeneous, then it can be transformed into a differential equation whose variables are separable b the substitution v where v is a differentiable function of EXAMPLE 6 Solving a Homogeneous Differential Equation Find the general solution of 2 2 d 3 d 0 STUDY TIP The substitution v will ield a differential equation that is separable with respect to the variables and v You must write our final solution, however, in terms of and Solution Because 2 2 and 3 are both homogeneous of degree 2, let v to obtain d dv v d Then, b substitution, ou have 2 v 2 2 d 3 v dv v d 0 2 2v 2 2 d 3 3 v dv 0 2 2v2 d 2 3v dv 0 Dividing b 2 and separating variables produces 2v 2 d 3v dv d 3v 2v dv 2 d ln 3 ln 2v2 C ln 3 ln ln ln 2v2 ln C C 2v2 3 C 2v 2 3 Substituting for v produces the following general solution C = C = C C 2 C General solution You can check this b differentiating and rewriting to get the original equation General solutions of 2 2 d 3 d 0 Figure 63 ( ) 3 = C 2 C = 3 C = Tr It TECHNOLOGY If ou have access to a graphing utilit, tr using it to graph several of the solutions in Eample 6 For instance, Figure 63 shows the graphs of C 2 for C, 2, 3, and Eploration A

5 SECTION 63 Separation of Variables and the Logistic Equation 25 Applications EXAMPLE 7 Wildlife Population The rate of change of the number of cootes N t in a population is directl proportional to 650 N t, where t is the time in ears When t 0, the population is 300, and when t 2, the population has increased to 500 Find the population when t 3 Solution Because the rate of change of the population is proportional to 650 N t, ou can write the following differential equation dn dt k 650 N You can solve this equation using separation of variables Differential form dn k dt Separate variables 650 N ln Integrate ln 650 N kt C 650 N kt C 650 N e kt C Assume N < 650 N 650 Ce kt General solution Using N 300 when t 0, ou can conclude that C 350, which produces N e kt dn k 650 N dt Then, using N 500 when t 2, it follows that e 2k e 2k 3 7 So, the model for the coote population is N e 0236t Model for population When t 3, ou can approimate the population to be N e cootes The model for the population is shown in Figure 6 k 0236 N 700 Number of cootes (0, 300) (2, 500) (3, 552) N = e 0236t Time (in ears) t Editable Graph Figure 6 Tr It Eploration A

6 26 CHAPTER 6 Differential Equations Each line K is an orthogonal trajector to the famil of circles Figure 65 A common problem in electrostatics, thermodnamics, and hdrodnamics involves finding a famil of curves, each of which is orthogonal to all members of a given famil of curves For eample, Figure 65 shows a famil of circles 2 2 C Famil of circles each of which intersects the lines in the famil K Famil of lines at right angles Two such families of curves are said to be mutuall orthogonal, and each curve in one of the families is called an orthogonal trajector of the other famil In electrostatics, lines of force are orthogonal to the equipotential curves In thermodnamics, the flow of heat across a plane surface is orthogonal to the isothermal curves In hdrodnamics, the flow (stream) lines are orthogonal trajectories of the velocit potential curves EXAMPLE Finding Orthogonal Trajectories Describe the orthogonal trajectories for the famil of curves given b C for C 0 Sketch several members of each famil Solution First, solve the given equation for C and write C Then, b differentiating implicitl with respect to, ou obtain the differential equation 0 d d d d Differential equation Slope of given famil Given famil: = C Orthogonal famil: 2 2 = K Because represents the slope of the given famil of curves at,, it follows that the orthogonal famil has the negative reciprocal slope So, d d Slope of orthogonal famil Now ou can find the orthogonal famil b separating variables and integrating d d Orthogonal trajectories Figure K 2 2 C The centers are at the origin, and the transverse aes are vertical for K > 0 and horizontal for K < 0 If k 0, the orthogonal trajectories are the lines ± If K 0, the orthogonal trajectories are hperbolas Several trajectories are shown in Figure 66 Tr It Eploration A

7 SECTION 63 Separation of Variables and the Logistic Equation 27 L = L Logistic curve Note that as t, L Figure 67 t Logistic Differential Equation In Section 62, the eponential growth model is derived from the fact that the rate of change of a variable is proportional to the value of You observed that the differential equation d dt k has the general solution Ce kt Eponential growth is unlimited, but when describing a population, there often eists some upper limit L past which growth cannot occur This upper limit L is called the carring capacit, which is the maimum population t that can be sustained or supported as time t increases A model that is often used for this tpe of growth is the logistic differential equation d dt k L Logistic differential equation where k and L are positive constants A population that satisfies this equation does not grow without bound, but approaches the carring capacit L as t increases From the equation, ou can see that if is between 0 and the carring capacit L, then d dt > 0, and the population increases If k is greater than L, then d dt < 0, and the population decreases The graph of the function is called the logistic curve, as shown in Figure 67 EXAMPLE 9 Deriving the General Solution EXPLORATION Use a graphing utilit to investigate the effects of the values of L, b, and k on the graph of L be kt Include some eamples to support our results Solve the logistic differential equation Solution Begin b separating variables L d kdt ln ln L Solving this equation for L produces be kt Write differential equation Separate variables Integrate each side Rewrite left side using partial fractions Find antiderivative of each side Multipl each side b and simplif Eponentiate each side Let ±e C b From Eample 9, ou can conclude that all solutions of the logistic differential equation are of the general form d kdt L L d kdt kt C ln L kt C L L L be kt d dt k L e kt C e C e kt be kt d dt k L Tr It Eploration A Eploration B

8 2 CHAPTER 6 Differential Equations EXAMPLE 0 Solving a Logistic Differential Equation A state game commission releases 0 elk into a game refuge After 5 ears, the elk population is 0 The commission believes that the environment can support no more than 000 elk The growth rate of the elk population p is dp dt kp 000 p, 0 p 000 where t is the number of ears a Write a model for the elk population in terms of t b Graph the slope field of the differential equation and the solution that passes through the point 0, 0 c Use the model to estimate the elk population after 5 ears d Find the limit of the model as t 5000 EXPLORATION Eplain what happens if p 0 L Solution a You know that L 000 So, the solution of the equation is of the form p 000 be kt Because p 0 0, ou can solve for b as shown be k b Then, because p 0 when t 5, ou can solve for k e k 5 b 99 k So, a model for the elk population is given b p 99e 09t b Using a graphing utilit, ou can graph the slope field of Figure 6 dp dt 09p 000 p and the solution that passes through 0, 0, as shown in Figure 6 c To estimate the elk population after 5 ears, substitute 5 for t in the model p e e Substitute 5 for t Simplif 000 d As t increases without bound, the denominator of gets closer to 99e 09t 000 So, lim t 09t e Tr It Eploration A

9 SECTION 63 Separation of Variables and the Logistic Equation 29 Eercises for Section 63 The smbol Click on Click on indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem to view the complete solution of the eercise to print an enlarged cop of the graph In Eercises, find the general solution of the differential equation d d 2 d 2 2 d dr dr 005r 005s ds ds ln 0 sin In Eercises 3 22, find the particular solution that satisfies the initial condition Differential Equation du 9 uv sin v2 dv dr 20 er 2s ds 2 dp kp dt 0 22 dt k T 70 dt 0 e 0 2 ln cos 3e 0 Initial Condition u 0 r 0 0 P 0 P 0 T 0 0 In Eercises 23 and 2, find an equation of the graph that passes through the point and has the given slope 3 f, 2 ln 32 f, tan 33 f, 2 ln 3 f, tan In Eercises 35 0, solve the homogeneous differential equation In Eercises, find the particular solution that satisfies the initial condition d 2e d d d 0 3 Differential Equation sec d d d d 0 Slope Fields In Eercises 5, sketch a few solutions of the differential equation on the slope field and then find the general solution analticall To print an enlarged cop of the graph, select the MathGraph button d d 5 6 d d Initial Condition ,,, 2, In Eercises 25 and 26, find all functions f having the indicated propert 25 The tangent to the graph of f at the point, intersects the -ais at 2, 0 26 All tangents to the graph of f pass through the origin 2 2 d 7 d 2 d 025 d In Eercises 27 3, determine whether the function is homogeneous, and if it is, determine its degree 27 f, f, f, f,

10 30 CHAPTER 6 Differential Equations Euler s Method In Eercises 9 52, (a) use Euler s Method with a step size of h 0 to approimate the particular solution of the initial value problem at the given -value, (b) find the eact solution of the differential equation analticall, and (c) compare the solutions at the given -value Differential Equation d d 6 d d 62 0 d 2 d 3 2 d d 2 2 Initial Condition 0, 5 0, 3, 2, 0 -value Radioactive Deca The rate of decomposition of radioactive radium is proportional to the amount present at an time The half-life of radioactive radium is 599 ears What percent of a present amount will remain after 25 ears? 5 Chemical Reaction In a chemical reaction, a certain compound changes into another compound at a rate proportional to the unchanged amount If initiall there are 20 grams of the original compound, and there is 6 grams after hour, when will 75 percent of the compound be changed? Slope Fields In Eercises 55 5, (a) write a differential equation for the statement, (b) match the differential equation with a possible slope field, and (c) verif our result b using a graphing utilit to graph a slope field for the differential equation [The slope fields are labeled (a), (b), (c), and (d)] To print an enlarged cop of the graph, select the MathGraph button 57 The rate of change of with respect to is proportional to the product of and the difference between and 5 The rate of change of with respect to is proportional to 2 59 Weight Gain A calf that weighs 60 pounds at birth gains weight at the rate dw dt where w is weight in pounds and t is time in ears Solve the differential equation (a) Use a computer algebra sstem to solve the differential equation for k 0, 09, and Graph the three solutions (b) If the animal is sold when its weight reaches 00 pounds, find the time of sale for each of the models in part (a) (c) What is the maimum weight of the animal for each of the models? 60 Weight Gain A calf that weighs w 0 pounds at birth gains weight at the rate dw dt k 00 w 00 w where w is weight in pounds and t is time in ears Solve the differential equation In Eercises 6 66, find the orthogonal trajectories of the famil Use a graphing utilit to graph several members of each famil C C 63 2 C 6 2 2C 65 2 C 3 66 Ce (a) 9 (b) 5 In Eercises 67 70, match the logistic equation with its graph [The graphs are labeled (a), (b), (c), and (d)] (a) (b) (c) (d) The rate of change of with respect to is proportional to the difference between and 56 The rate of change of with respect to is proportional to the difference between and (c) (d)

11 SECTION 63 Separation of Variables and the Logistic Equation 3 67 e e 70 In Eercises 7 and 72, the logistic equation models the growth of a population Use the equation to (a) find the value of k, (b) find the carring capacit, (c) find the initial population, (d) determine when the population will reach 50% of its carring capacit, and (e) write a logistic differential equation that has the solution P t P t 72 P t 2e 075t 39e 02t In Eercises 73 and 7, the logistic differential equation models the growth rate of a population Use the equation to (a) find the value of k, (b) find the carring capacit, (c) use a computer algebra sstem to graph a slope field, and (d) determine the value of P at which the population growth rate is the greatest dp dp P 0000P2 dt 3P 00 P dt In Eercises 75 7, find the logistic equation that satisfies the initial condition Logistic Differential Equation d dt 0 d dt d 2 dt 5 50 d 3 2 dt e e 2 Initial Condition 0, 0, 5 0, 0, 5 79 Endangered Species A conservation organization releases 25 Florida panthers into a game preserve After 2 ears, there are 39 panthers in the preserve The Florida preserve has a carring capacit of 200 panthers (a) Write a logistic equation that models the population of the panther population of the preserve (b) Find the population of the herd after 5 ears (c) When will the herd s population reach 00? (d) Write a logistic differential equation that models the growth rate of the panther population Then repeat part (b) using Euler s Method with a step size of h Compare the approimation with the eact answers (e) At what time is the panther population growing most rapidl? Eplain 0 Bacteria Growth At time t 0, a bacterial culture weighs gram Two hours later, the culture weighs 2 grams The maimum weight of the culture is 0 grams (a) Write a logistic equation that models the weight of the bacterial culture (b) Find the culture s weight after 5 hours (c) When will the culture s weight reach grams? (d) Write a logistic differential equation that models the growth rate of the culture s weight Then repeat part (b) using Euler s Method with a step size of h Compare the approimation with the eact answers (e) At what time is the culture s weight increasing most rapidl? Eplain Writing About Concepts In our own words, describe how to recognize and solve differential equations that can be solved b separation of variables 2 State the test for determining if a differential equation is homogeneous Give an eample 3 In our own words, describe the relationship between two families of curves that are mutuall orthogonal Sailing Ignoring resistance, a sailboat starting from rest accelerates dv dt at a rate proportional to the difference between the velocities of the wind and the boat (a) The wind is blowing at 20 knots, and after minute the boat is moving at 5 knots Write the velocit v as a function of time t (b) Use the result of part (a) to write the distance traveled b the boat as a function of time True or False? In Eercises 5, determine whether the statement is true or false If it is false, eplain wh or give an eample that shows it is false 5 The function 0 is alwas a solution of a differential equation that can be solved b separation of variables 6 The differential equation can be written in separated variables form 7 The function f, 2 2 is homogeneous The families 2 2 2C and 2 2 2K are mutuall orthogonal Show that if then d k be kt, dt Putnam Eam Challenge 90 A not uncommon calculus mistake is to believe that the product rule for derivatives sas that fg f g If f e 2, determine, with proof, whether there eists an open interval a, b and a nonzero function g defined on a, b such that this wrong product rule is true for in a, b This problem was composed b the Committee on the Putnam Prize Competition The Mathematical Association of America All rights reserved