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Unformatted text preview: 502 CHAPTER 8 TECHNIQUES OF INTEGRATION Multiplying the differential equation by 9 + t , we have (9 + t ) dP + P = 240(9 + t ), dt and d [(9 + t )P] = 240(9 + t ), dt (9 + t )P = 120(9 + t )2 + C , C P (t ) = 120(9 + t ) + . 9+t Since the amount of pollutant in the tank is initially 2 × 360 = 720 (pounds), we see that P (0) = 120(9) + Thus, the function P (t ) = 120(9 + t ) − 3240 9+t (pounds). C = 720 9 which implies that C = −3240. gives the amount of pollutant in the tank at any time t . After 10 hours, there are 360 + 40(10) = 760 gallons of ﬂuid in the tank, and there are P (10) = 120(19) −
3240 19 ∼ 2109 = pounds of pollutant. Therefore, the rate at which pollutant is being released into the sewage system after 10 hours is 2109 ∼ 2. 78 pounds per gallon. 760 = EXERCISES *8.8
Determine whether the functions satisfy the differential equation. 1. 2y − y = 0; 2. y + xy = x; 3. y 4. y 5. y 6. y
2 y 1 ( x ) = e x /2 , y1 ( x ) = e
−x2 /2 y2 (x) = x2 + 2ex/2 . , y2 (x) = 1 + Ce−x
2 /2 , 2 y = 0. x+1 2 y = (x + 1)5/2 . 22. y + x+1 21. y + Find the particular solution determined by the initial condition. 23. y + y = x, 24. y − y = e ,
2x 1 1 + y = y ; y1 (x) = x , y2 ( x ) = , e +1 C ex + 1 + 4y = 0; y1 (x) = 2 sin 2x, y2 (x) = 2 cos x. − 4y = 0; y1 (x) = e2x , y2 (x) = C sinh 2x, − 2y − 3y = 7e3x ; y1 (x) = e−x + 2e3x , y2 (x) = 7 xe3x . 4 8. xy − 2y = −x. 10. y − y = −2 e−x . cos x 12. xy + 2y = . x 14. y + y = 2 + 2x. 16. y − y = ex . y(0) = 1. y(1) = 1. Find the general solution. 7. y − 2y = 1. 9. 2y + 5y = 2. 11. y − 2y = 1 − 2x. 13. xy − 4y = −2nx. 15. y − ex y = 0. 17. (1 + ex ) y + y = 1. 18. xy + y = (1 + x) ex . 19. y + 2xy = x e−x .
2 1 , y(0) = e. 1 + ex 1 26. y + y = , y(0) = e. 1 + 2 ex 25. y + y = 27. xy − 2y = x3 ex , 28. xy + 2y = x e ,
−x y(1) = 0. y(1) = −1. 20. xy − y = 2x ln x. 29. Find all functions that satisfy the differential equation y − y = y − y . HINT: Set z = y − y. 30. Find the general solution of y + ry = 0 on [0, ∞) where r is a constant. (a) Show that if y is a solution and y(a) = 0 at some number a ≥ 0, then y(x) = 0 for all x. (A solution y is either identically zero or never zero.) ∗ 8.8 DIFFERENTIAL EQUATIONS; FIRST-ORDER LINEAR EQUATIONS 503 (b) Show that if r < 0, then all nonzero solutions are unbounded. (c) Show that if r > 0, then all solutions have limit 0 as x → ∞. (d) Describe all solutions of the equation in the case r = 0. Exercises 31 and 32 are concerned with the ﬁrst-order linear differential equation (∗) y + p(x)y = 0 Here we are measuring distance in feet and the positive direction is down. (a) Find v(t ). (b) Show that v(t ) cannot exceed 32/k and that v(t ) → 32/k as t → ∞. (c) Sketch the graph of v(t ). 38. Suppose that a certain population P has a known birth rate d B/dt and a known death rate dD/dt . Then the rate of change of P is given by dB dD dP = − . dt dt dt (a) Assume that dB/dt = aP and dD/dt = bP , where a and b are constants. Find P (t ) if P (0) = P0 > 0. (b) Analyze the cases (i) a > b, (ii) a = b, (iii) a < b. Find lim P (t ) in each case.
t →∞ where the function p is continuous on an interval I . 31. (a) Show that if y1 and y2 are solutions of (∗), then u = y1 + y2 is also a solution of (∗). (b) Show that if y is a solution of (∗) and C is a constant, then u = Cy is also a solution of (∗). 32. (a) Let a ∈ I . Show that the general solution of (∗) can be written as y(x) = Ce−
x a p(t ) dt . (b) Show that if y is a solution of (∗) and y(b) = 0 for some b ∈ I , then y(x) = 0 for all x ∈ I . (c) Show that if y1 and y2 are solutions of (∗) and y1 (b) = y2 (b) for some b ∈ I , then y1 (x) = y2 (x) for all x ∈ I . Exercises 33 and 34 are concerned with the differential equation (8.8.1): y + p(x)y = q(x) where p and q are functions continuous on some interval I . 33. Let a ∈ I and let H (x) =
x a 39. The current i in an electrical circuit consisting of a resistance R, inductance L, and voltage E varies with time (measured in seconds) according to the formula L di + Ri = E , dt R > 0, L > 0, E > 0 constant. p(t ) dt . Show that
x (a) Find i if i(0) = 0. (b) What upper limit does the current approach as t → ∞? (c) In how many seconds will the current reach 90% of its limit? 40. The current i in an electrical circuit consisting of a resistance R, inductance L, and a voltage E sin ωt , varies with time according to the formula L di + Ri = E sin ωt , dt R > 0, L > 0, E > 0 constant y(x) = e−H (x)
a q(t ) eH (t ) dt 34. 35. 36. 37. is the solution of the differential equation that satisﬁes the initial condition y(a) = 0. Show that if y1 and y2 are solutions of (8.8.1), then z = y1 − y2 is a solution of the differential equation (∗) above. A thermometer is taken from a room where the temperature is 72◦ F to the outside, where the temperature is 32◦ F. After 1 minute the thermometer reads 50◦ F. What will the ther2 mometer read at t = 1 minute? How long will it take for the thermometer to read 35◦ ? A solid metal sphere at room temperature of 20◦ C is dropped into a container of boiling water (100◦ C ). If the temperature of the sphere increases 2◦ in 2 seconds, what will the temperature be at time t = 6 seconds? How long will it take for the temperature of the sphere to reach 90◦ C ? An object falling from rest in air is subject not only to the gravitational force but also to air resistance. Assume that the air resistance is proportional to the velocity and acts in a direction opposite to the motion. Then the velocity of the object at time t satisﬁes v = 32 − kv(t ), k > 0 constant , v(0) = 0. (a) Find i if i(0) = i0 . (b) Does lim i(t ) exist? Compare with Exercise 39.
t →∞ (c) Sketch the graph of i. 41. A 200-liter tank initially full of water develops a leak at the bottom. Given that 20% of the water leaks out in the ﬁrst 5 minutes, ﬁnd the amount of water left in the tank t minutes after the leak develops: (a) if the water drains off at a rate that is proportional to the amount of water present. (b) if the water drains off at a rate that is proportional to the product of the time elapsed and the amount of water present. 42. At a certain moment a 100-gallon mixing tank is full of brine containing 0.25 pounds of salt per gallon. Find the amount of salt present t minutes later if the brine is being continuously drawn off at the rate of 3 gallons per minute and replaced by brine containing 0.2 pounds of salt per gallon. 43. An advertising company is trying to expose a new product to a certain metropolitan area by advertising on television. 504 CHAPTER 8 TECHNIQUES OF INTEGRATION Suppose that the exposure to new people is proportional to the number of people who have not seen the product out of a total population of M viewers. Let P (t ) denote the number of viewers who have been exposed to the product at time t . The company has determined that no one was aware of the product at the start of the campaign [P (0) = 0] and that 30% of the viewers were aware of the product after 10 days. (a) Determine the differential equation that describes the number of viewers who are aware of the product at time t . (b) Determine the solution of the differential equation from part (a) that satisﬁes the initial condition P (0) = 0. (c) How long will it take for 90% of the population to be aware of the product? 44. A drug is fed intravenously into a patient’s bloodstream at the constant rate r . Simultaneously, the drug diffuses into the patient’s body at a rate proportional to the amount of drug present. (a) Determine the differential equation that describes the amount Q(t ) of the drug in the patient’s bloodstream at time t . (b) Determine the solution Q = Q(t ) of the differential equation found in part (a) that satisﬁes the initial condition Q(0) = 0. (c) Find lim Q(t ).
t →∞ describes a population that undergoes periodic ﬂuctuations. Assume that P (0) = 1000 and ﬁnd P (t ). Use a graphing utility to draw the graph of P . (b) The differential equation dP = (2 cos 2π t )P + 2000 cos 2π t dt describes a population that undergoes periodic ﬂuctuations as well as periodic migration. Continue to assume that P (0) = 1000 and ﬁnd P (t ) in this case. Use a graphing utility to draw the graph of P and estimate the maximum value of P . c 46. The Gompertz equation dP = P (a − b ln P ), dt where a and b are positive constants, is another model of population growth. (a) Find the solution of this differential equation that satisﬁes the initial condition P (0) = P0 , where P0 > 0. HINT: Deﬁne a new dependent variable Q by Q = ln P . (b) Find lim P (t ).
t →∞ c 45. (a) The differential equation
dP = (2 cos 2π t )P dt (c) Determine the concavity of the graph of P . (d) Use a graphing utility to draw the graph of P in the case where a = 4, b = 2, and P0 = 1 e2 . Does the graph 2 conﬁrm your result in part (c)? ∗ 8.9 INTEGRAL CURVES; SEPARABLE EQUATIONS Integral Curves
If a function y = y(x) satisﬁes a ﬁrst-order differential equation, then along the graph of the function, the numbers x, y, y are related as prescribed by the equation. In the case of a nonlinear differential equation, it is usually difﬁcult (and sometimes impossible) to ﬁnd functions which are explicitly deﬁned and satisfy the differential equation. More often, we can ﬁnd plane curves which, though not the graphs of functions, do have the property that along the curve the numbers x, y, y are related as prescribed by the differential equation. Such curves are called integral curves (solution curves) of the differential equation. To solve a nonlinear differential equation is to ﬁnd the integral curves of the equation. Separable Equations
A ﬁrst-order differential equation is said to be separable if it can be written in the form
(8.9.1) p(x) + q(y)y = 0. The functions p and q are assumed to be continuous where deﬁned. ...
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