Differential Equations

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1 Differential Equations Big Ideas Slope fields draw a slope field, sketch a particular solution Separation of variables separable differential equations General solution Particular solution Growth decay problems Antidifferentiation by substitution Antidifferentiation by parts Winplot graphing program is very good for drawing slope fields. Domain Issues The Domain of Solutions to Differential Equations by Larry Riddle (AP Central) Definition: The solution of a differential equation is a differentiable function on an open interval that contains the initial -value. For all parts of the domain, the derivative of the eplicit solution does not contradict the original differential equation The derivative eists for all values in its domain Mike Koehler 6 - Differential Equations

2 Mike Koehler 6 - Differential Equations

3 Remember the Domain!. Find a solution y = f( ) to the differential equation d = satisfying y( ) =.. Find a solution y = f( ) to the differential equation = y satisfying () d y =.. Find a solution y = f( ) to the differential equation = y satisfying () 0 d y =.. Find a solution y f( ) = to the differential equation = ( y + ) satisfying (0) d y =. Mike Koehler 6 - Differential Equations

4 Mike Koehler 6 - Differential Equations

5 AP Multiple Choice Questions 008 AB Multiple Choice BC Multiple Choice 7 00 AB Multiple Choice. The rate of change of the volume,v, of water in a tank with respect to time, t, is directly proportional to the square root of the volume. Which of the following is a differential equation that describes this relationship? A) Vt () = k t B) Vt () = k V C) dv dv k = k t D) = dt dt V E) dv = k V dt 9. A curve has slope + at each point (, ) curve if it passes through the point (, )? A) y = 5 B) D) y = + E) y on the curve. Which of the following is an equation for this y y = + = + C) y = + 00 BC Multiple Choice. Shown to the right is a slope field for which of the following differential equations? A) = B) d y = C) d y = d y D) = E) d y = d y Mike Koehler 6-5 Differential Equations

6 998 AB Multiple Choice. If ky and k dt = is a nonzero constant, then y could be A) e kty B) e kt kt C) e + D) kty + 5 E) ky + 8. Population y grows according to the equation ky d =, where k is a constant and t is measured in years. If the population doubles every 0 years, then the value of k is A) B).000 C) 0.0 D). E) BC Multiple Choice 8. If sin( )cos ( ) d = and if 0 when π y = =, what is the value of y when = 0? A) - B) C) 0 D) E) 99 AB Multiple Choice. If y and if y when, then when, y d = = = = = A) B) C) 0 D) E). A puppy weighs.0 pounds at birth and.5 pounds two months later. If the weight of the puppy during its first 6 months is increasing at a rate proportional to its weight, then how much will the puppy weigh when it is months old? A). pounds B).6 pounds C).8 pounds D) 5.6 pounds E) 6.5 pounds 99 BC Multiple Choice. If y, then ycould be d = A) ln B) 7 e + C) e D) e E) + 8. During a certain epidemic, the number of people that are infected at any time increases at a rate proportional to the number of people that are infected at that time. If,000 people are infected when the epidemic is first discovered, and,00 are infected 7 days later, how many people are infected days after the epidemic is first discovered? A) B), C),67 D),00 E),057 Mike Koehler 6-6 Differential Equations

7 988 BC Multiple Choice 9. If ysec and y 5 when 0, then y d = = = = tan( ) A) tan( ) tan( ) e + B) e + 5 C) 5e D) tan( ) + 5 E) tan( ) + 5e. Bacteria in a certain culture increase at a rate proportional to the number present. If the number of bacteria doubles in three hours, in how many hours will the number of bacteria triple? ln ln ln 7 A) B) C) D) ln ln ln ln 9 E) ln 985 BC Multiple Choice. At each point ( y, ) on a certain curve, the slope of the curve is y. If the curve contains the point ( 0,8 ), then its equation is A) y = 8e B) y = + 8 C) y = e + 7 D) y ln ( ) 8 y = + 8 = + + E) Integration by substitution Multiple Choice 00 ab 0 A) e d = e B) e C) e D) e E) e 00 ab Using the substitution u = +, + d is equivalent to A) D) u du B) u du E) 0 0 u du C) 0 5 u du 5 u du 998 7ab What is the average value of A) 6 9 B) 5 9 = + on the interval [ ] y C) 6 0,? D) 5 E) Mike Koehler 6-7 Differential Equations

8 998 8ab If f is a continuous function and if F'( ) = f( ) for all real numbers, then A) F() F() B) F() F() C) F(6) F() D) F(6) F() E) F(6) F() f ( ) d = 997 6ab t e dt = t A) e B) e t t C) e D) e t t E) e 997 8ab π tan e d = 0 cos A) 0 B) C) e D) e E) e bc 0 e d = A) ( ) e B) e C) e D) e E) ( e ) 988 7ab d = A) ( ) + B) ( ) + 5 C) ( ) 5 D) ( ) + E) ( ) ab tan ( ) d = ln cos A) ln cos( ) B) ( ) C) ( ) D) ln cos( ) ln cos E) sec ( ) tan ( ) 985 ab π 0 ( ) sin d = A) - B) C) 0 D) E) Mike Koehler 6-8 Differential Equations

9 985 bc + d = + A) ln 8 ln B) ln 8 ln C) ln 8 D) ln E) ln bc If the substitution u = is made, the integral d = A) u u du B) du u u C) D) u u du E) u du u u u du 97 ab 0 ( ) + + e d = e A) B) e C) e e D) e E) e e 97 7ab 0 d = A) B) ln C) π 6 π D) E) bc 969 ab d = A) ( ) B) ( ) C) ( ) D) ( ) E) ( ) ( ) sin + d = cos cos A) ( + ) B) cos( + ) C) cos( ) cos + 5 D) ( + ) E) ( ) + Mike Koehler 6-9 Differential Equations

10 Integration by parts Multiple Choice 00 bc sin(6 ) d = A) cos(6 ) + sin(6 ) B) C) cos(6 ) + sin(6 ) 6 6 D) E) 6cos(6 ) sin(6 ) cos(6 ) + sin(6 ) 6 6 cos(6 ) + sin(6 ) ab f ( ) d = A) f( ) f ( ) d B) D) f ( ) f ( ) d E) f ( ) f ( ) d C) f ( ) d = f( ) f( ) 997 5ab sin ( ) d = cos sin C cos sin A) cos( ) + sin ( ) B) cos( ) sin ( ) C) ( ) ( ) + D) cos( ) sin ( ) E) ( ) ( ) bc sin d = A) cos sin cos B) cos + sin cos + C C) cos + sin + cos D) E) cos + C cos + C 988 6bc e d = A) D) e e B) e e + E) e e C) e e e bc If f ( )sin d = f ( )cos + cos d, then f ( ) could be A) B) C) D) sin E) cos (insightful) Mike Koehler 6-0 Differential Equations

11 AP Free Response Questions 0 AB5 The rate at which a baby bird gains weight is proportional to the difference between its adult weight and its current weight. At time t = 0, when the bird is first weighed, its weight is 0 grams. If Bt () is the weight of the bird, in grams, at time t days after it is first weighed, then db ( 00 B) dt = 5. Let y= Bt () be the solution to the differential equation above with initial condition B (0) = 0. a) Is the bird gaining weight faster when it weighs 0 grams or when it weighs 70 grams? Eplain your reasoning. b) d B d B Find in terms of B. Use to eplain why the graph of B cannot resemble the graph above. dt dt c) Use separation of variables to find y= Bt (), the particular solution to the differential equation with initial condition B (0) = 0. 0 AB5 At the beginning of 00, a landfill contained 00 tons of solid waste. The increasing function W models the total amount of solid waste stored at the landfill. Planners estimate that W will satisfy the differential equation dw = ( W 00) for the net 0 years. W is measured in tons, and t is measured in years from the start of 00. dt 5 a) Use the line tangent to the graph of W at t = 0 to approimate the amount of solid waste that the landfill contains at the end of the first months of 00 time t =. b) dw dw Find in terms of W. Use to determine whether your answer in part (a) is an underestimate or dt dt an overestimate of the amount of solid waste that the landfill contains at time t =. c) dw Find the particular solution W= Wt () to the differential equation = ( W 00) with initial condition dt 5 W (0) = 00. Mike Koehler 6 - Differential Equations

12 00 AB6 d y Solutions to the differential equation = y also satisfy = y ( + y ). Let y = f( ) be a particular d d solution to the differential equation = y with () d f =. a) Write an equation for the line tangent to the graph of y = f( ) at =. b) Use the tangent line equation from part (a) to approimate f (.). Given that f( ) > 0 for < <., is the approimation for f (.) greater than or less than f (.)? Eplain your reasoning. c) Find the particular solution y = f( ) with initial condition f () =. 008 AB5 y Consider the differential equation =, where 0. d a) On the aes provided, sketch a slope field for the given differential equation at the nine points indicated. b) Find the particular solution y = f( ) to the differential equation with the initial condition f () = 0. c) For the particular solution y = f( ) described in part (b), find lim f( ) 006 AB 5 Consider the differential equation + y =, where 0. d a) On the ais provided, sketch a slope field for the given differential equation at the eight points indicated. b) Find the particular solution y = f( ) to the differential equation with the initial condition f ( ) = and state its domain. Mike Koehler 6 - Differential Equations

13 005 AB 6 Consider the differential equation =. d y a) On the ais provided, sketch a slope field for the given differential equation at the twelve points indicated. b) Let y = f( ) be the particular solution to the differential equation with the initial condition f () =. Write an equation for the line tangent to the graph of f at (, ) and use it to approimate f (.). c) Find the particular solution y = f( ) to the given differential equation with the initial condition f () =. 00 AB6 Consider the differential equation ( y ) d =. a) On the ais provided, sketch a slope field for the given differential equation at the twelve points indicated. b) While the slope field in part (a) is drawn for only twelve points, it is defined at every point in the y-plane. Describe all points in the y-plane for which the slopes are positive. c) Find the particular solution y = f( ) to the given differential equation with the initial condition f (0) =. Mike Koehler 6 - Differential Equations

14 00 AB5 A coffeepot has the shape of a cylinder with radius 5 inches. Let h be the depth of the coffee in the pot, measured in inches, where h is a function of time t, measured in seconds. The volume V of coffee in the pot is changing at the rate of 5π h cubic inches per second. (The volume V of a cylinder with radius r and height h is V= π rh.) a) dh h Show that dt = 5. b) dh h Given that h = 7 at time t = 0 solve the differential equation = for h as a function of t. dt 5 c) At what time t is the coffeepot empty? 000 AB6 Consider the differential equation =. y d e a) Find a solution y = f( ) to the differential equation satisfying f (0) =. b) Find the domain and range of the function f found in part (a). 998 AB Let f be a function with f () = such that for all points ( yon, ) the graph of f the slope is given by a) Find the slope of the graph of f at the point where =. b) Write an equation for the line tangent to the graph of f at = and use it to approimate f (.). c) Find f( ) by solving the separable differential equation d) Use your solution from part c to find f (.). + = with initial condition f () = d y +. y 99 AB 6 Let Pt () represent the number of wolves in a population at time t years, when t 0. The population Pt () is increasing at a rate directly proportional to 800 Pt ( ), where the constant of proportionality is k. a) If P(0) = 500, find Pt ( ) in term of tand k. b) If P() = 700, find k. c) Find lim Pt ( ). t Mike Koehler 6 - Differential Equations

15 989 AB6 Oil is being pumped continuously from a certain oil well at a rate proportional to the amount of oil left in the well; that is, ky dt =, where y is the amount of oil left in the well at any time t. Initially there were,000,000 gallons of oil in the well, and 6 years later there were 500,000 gallons remaining. It will no longer be profitable to pump oil when there are fewer than 50,000 gallons remaining. a) Write an equation for y, the amount of oil remaining in the well at any time t. b) At what rate is the amount of oil in the well decreasing when there are 600,000 gallons of oil remaining? c) In order not to lose money, at what time t should oil no longer be pumped from the well? 000 BC6 y d =. Consider the differential equation given by ( ) a) On the ais provided, sketch a slope field for the differential equation at the eleven points indicated. b) Use the slope field for the given differential equation to eplain why a solution could not have the graph shown in the figure on the right above. c) Find the particular solution y = f( ) to the given differential equation with the initial condition f (0) =. d) Find the range of the solution fount in part (c). Mike Koehler 6-5 Differential Equations

16 Mike Koehler 6-6 Differential Equations

17 Tetbook Problems Calculus, Finney, Demanna, Waits, Kenne; Prentice Hal, l0 Section Questions QQ p QQ p76 7.R 9-60 Handout Problems Mike Koehler 6-7 Differential Equations

18 Mike Koehler 6-8 Differential Equations

19 AP Calculus Chapter 6 Section Slope Fields A slope field is a plot of short line segments with slope f( y, ) for a lattice of points in the plane. Slope fields enable us to graph solution curves without solving the differential equation. Let = f (, y) = y + y. Sketch the slope field at the points indicated on the ais provided. d For the point (,), the slope is equal to + =. Draw a short line segment with slope through the point (,). Repeat for each of the lattice points in the graph below. Draw a possible graph for the function f with the given slope field that goes through the point (0,). Mike Koehler 6-9 Differential Equations

20 Mike Koehler 6-0 Differential Equations

21 AP Calculus Chapter 6 Section Slope Fields Draw the slope field for each of the following differential equations.. d = +. y d =. y d = +. d = 5. y d = 6. y = d Mike Koehler 6 - Differential Equations

22 Match the following differential equations to their slope fields. i..5y d = + ii. = d y iii. d = iv. y d = v..5y d = A. B C. D E. - Mike Koehler 6 - Differential Equations

23 AP Calculus Chapter 6 Section Match the following integrals to one of the following types: A) Identify u for each integral. du B) u n u du C) u e du D) Other. e d u =. d u = + d. u =. ln ln d u = ln d u = 6. d u = + d u = 8. d u = + 5 e 9. tan sec d u = 0. e d u = + e. ( + ) d u =. d u =. 5. e d u =. d u = 6. + e e + e e d u = sin cos ( e ) d u = tan 7. d u = 8. cos sin cos d u = 9. sin d u = 0. + cos 6 ln d u =. cos( ) d u =. + d u =. sin d u =. ln(cos ) tan d u = e 5. d u = 6. (e 5) e d u = ( e + ) Mike Koehler 6 - Differential Equations

24 AP Calculus Chapter 6 Section Answers Match the following integrals to one of the following types: A) Identify u for each integral. du B) u n u du C) u e du D) Other.C e d u =.A d u = + +.A d ln u = ln.b ln d u = ln 5.B ln d u ln = 6.B d u = 7.B d u = C d e u = 9.B tan sec d u tan = 0.A e d u = + e + e.b ( + ) d u = +.A d u =.B e d u =.A e e + e e d u = e e 5.B d u = + 6.C + sin cos ( e ) d u = sin tan 7.B d u = tan 8.B cos sin cos d u = cos 9.A sin d u = + cos 0.B + cos 6 ln d u = ln.d cos( ) d u =.B + d u = +.B sin d u sin =.B ln(cos ) tan d u = ln(cos ) e 5.B d u = e 5 6.A (e 5) e d u = e + ( e + ) Mike Koehler 6 - Differential Equations

25 AP Calculus Chapter 6 Section Part : Calculate du for the given function.. u =. u = sin. u = cos( ) 6. tan u = + u = u = Part : Write the integral in terms of u and du. Then evaluate the integrals. Final answer should be in terms of.. ( 7) d u = 7. + d u = +. ( + ) d u = +. ( + ) d u = + 5. sin( ) d u = 6. d u = + 9 ( + ) + 7. d u = + 8. d u = sin d u = ( + ) ( 8 + 5) ( ) ( ) 0. d u =. d u =. + sin 9 cos d u = + sin. cos( ) d u =. sin ( )cos( ) d u = sin( ) 5. sec ( ) tan( ) d u = tan( ) Part : Evaluate the indefinite integrals.. ( + ) d. ( + ) d. d 7 ( ) ( + )( + ). sin 7 d 5. d 6. d + 7. d 8. d 9. d ( ) ( + ) d. d. + d ( )( ) 5 ( + ) ( ) ( + ) ( + ) ( + ) d. 9 d 5. d 5 ( ) d 7. sin ( ) cos( ) 8. sin ( + ) 6. sin d d ( ) ( ) ( ) 9. sec + 9 d 0. sec tan d. sin cos + d cos( ) ( ) ( + sin ) cos. d. cos sin d. d Mike Koehler 6-5 Differential Equations

26 AP Calculus Chapter 6 Section Answers Part :. du = d. du = cosd. du = d ( ) sin( ) 6. sec du = + d du = d du = d Part :. u du = ( 7 ). u du = ( + ). u du = ( u ) u du = u u du = ( + ) ( + ) 0 5. sin udu = cos( ) 6. u du ( ) C = + + u C du = du u u du u = = ( + ) u ( 8 5) ( 8 5) 9. sin udu = cos u 5 0. ( ). ( ) ( ) u du = 6 u du = 6 u u du = + C u du = ( + sin ) 0. cos( u) du = sin ( ). u du = sin ( ) 5. u d tan u = + C Part : 5. ( + ). ( + ) cos( 7) 5. ( ) 6. ( + ) ( + ) ( ) 5 0. ( + ).. ( + ) 5 5 ( + ). 5. ( + 9). ( + 9) ( + 9) ( + ) + ( + ) + ( + ) cos( ) 7. sin ( ) 8. cos( + ) tan ( + 9) 0. tan ( ). ( cos + ) 5 6. ( + sin ). ( sin ) 6. sin Mike Koehler 6-6 Differential Equations

27 AP Calculus Chapter 6 Section. Consider the differential equation =. d y a. Let y = f( ) be the particular solution to the given differential equation for < < 5such that the line y = is tangent to the graph of f. Find the -coordinate of the point of tangency, and determine whether f has a local maimum, local minimum, or neither at this point. Justify your answer. b. Let y= g ( ) be the particular solution to the given differential equation for < < 8, with the initial condition g (6) =. Find y= g ( ).. Let f be the function satisfying f ( ) = f( ) for all real numbers, where f () = 5. a. Find f (). b. Write an epression for y = f( ) by solving the differential equation y d = with the initial condition f () = 5 =. d. Consider the differential equation ( y ) cos( π ) a. On the aes provided, sketch a slope field for the given differential equation at the nine points indicated. b. There is a horizontal line with equation y = c that satisfies this differential equation. Find the value of c. c. Find the particular solution y = f( ) to the differential equation with the initial condition f () = 0. Show the work that leads to your answer. Mike Koehler 6-7 Differential Equations

28 AP Calculus Chapter 6 Section Answers. a. 0 when. d = = Use second derivative test or first derivative test to justify that the function has a minimum at =. b. y = a. f ( ) = f( ) +. Show the work that leads to this answer. 9 f () =. b. y = + = ( + ) 6. a. b. The line y = satisfies the differential equation so c =. c. y = π sin for - + π < < ( π) Mike Koehler 6-8 Differential Equations

29 AP Calculus Chapter 6. Shown on the right is a slope field for which of the following differential equations? A) D) y d = B) y y d = C) y y d = + = y + E) ( ) d d = +. d =. cos( ) d =. e d = 5. Let R be the region between y = e and the -ais for. Find the area of R. Solve analytically. 6. Solve the differential equation = with initial condition y () =. d y 7. The rate of change of the volume, V, of water in a tank with respect to time, t, is directly proportional to the square root of the volume. Write a differential equation that describes this relationship. 8. The slope of the line tangent to the curve y = f( ) is given by point (, ), find the positive value of when y =. y d =. If the curve passes through the 9. Solve the differential equation = y and () d y =. Mike Koehler 6-9 Differential Equations

30 0. At any time t 0, in days, the rate of growth of a bacteria population is given by ky dt =, where k is a constant and y is the number of bacteria present. The initial population is 000 and the population triples during the first 5 days. a. Write an epression for y at any time t 0. b. What will the population of bacteria be in days? c. When will there be 6000 bacteria?. Consider the differential equation ( y ) d = a. On the aes provided, sketch a slope field for the given differential equation. b. While the slope field in part (a) is drawn at only twelve points, it is defined at every point in the y-plane. Describe all points in the y-plane for which the slopes are negative. c. Find the particular solution y = f( ) to the given differential equation with the initial condition f (0) = 0. Answers. C. ln. sin ( ). e 6 8 e 5. ( e e ) = 8 e y = dv = k V dt 8. 8 = 9. y = e ln t t 0. a. y = 000e = 000e b. 967 bacteria 5ln(6) c. t = = 8.56 days ln(). a. Slope field. b. Slopes are negative at points ( y, ) where 0 and y<. c. 5 5 y = e Mike Koehler 6-0 Differential Equations

( + ) 3. AP Calculus BC Chapter 6 AP Exam Problems. Antiderivatives. + + x + C. 2. If the second derivative of f is given by f ( x) = 2x cosx

( + ) 3. AP Calculus BC Chapter 6 AP Exam Problems. Antiderivatives. + + x + C. 2. If the second derivative of f is given by f ( x) = 2x cosx Chapter 6 AP Eam Problems Antiderivatives. ( ) + d = ( + ) + 5 + + 5 ( + ) 6 ( + ). If the second derivative of f is given by f ( ) = cos, which of the following could be f( )? + cos + cos + + cos + sin