7.2009 Differential Equation

Differential Equations In this chapter, students are i) expected to formulate a simple statement involving a rate of change as a differential equation, including the introduction of a constant of proportionality. ii) to find a general form of solution for a differential equation in which the variables are separable. iii) to sketch family of curves iv) to use an initial condition to find a particular solution of a differential equation. Differential Equations of Separable Variable dy = tan y cos x, given that when 1. Solve the differential equation dx x=

π 4

,y=

π 4

dy = xe y (where x ≥ 0) and if y =0 when x = 0, prove that y = - 2. 2. If dx dy = xy given that y = 2 when x = 0. 3. Solve the differential equation dx dy = y cot x, (0 < x < π, y > 0), 4. (a) Solve the differential equation dx π given that y = 2 when x = . 6 dy = 1 – y, given that y < 1 and (b) Solve the differential equation dx that y = 0 when x = 0. (c) Find the general solution of the differential equation dy cos2x = 1 – y2, where 0 < y< 1. dx dy = (x + 1)(y + 1), (y > -1), and that 5. Find y in terms of x given that dx y = 0 when x = 0. dx k = , where k is constant, find an expression for x n terms of t, dt xt given that x = 2 when t = 1.

6.

If

7.

Find y in terms of x given that (x2 + x)

8.

Solve the differential equation cos θ

dy = dx

y ,(x > 0, y > 0),

dr + r sin θ = 0, given that dθ r = a when θ = 0, expressing r in terms of a and θ.

1

9.

Two variables x and t are connected by the differential equation dx kx , where 0 < x < 10 and where k is a constant. It is given = dt 10 − x that x = 1 when t = 0 and that x = 2 when t = 1. Find the value of t when x = 5, giving three significant figures in your answer. dy 2y − = 0. dx x Sketch the two solution curves passing respectively through the points (2, 2) and (-1, -1).

10.Find the general solution of the differential equation

Answers: ln sin y = sin x + ln

1.

1

3. y = 2e 2 x

2

1 2

−

4. a) y = 4 sin x

1 2

1

2. ln(1 -

3

3

x 2)

1+ y   = 2 tan x + c) ln  1− y 

b) y = 1 − e − x

c 1

5. y= e 2 x

2

+x

−1

6. x =

8. r = a cos θ 9. 2.04

2 k ln t + 4

7. y= ¼ (ln

Ax x +1

)2

10. y = A x2

Expressing statements in the form of a differential equation 1 It is known that the rate of decay of a radioactive substance is proportional to the amount present. Let y be the amount of the radioactive substance present at time t. 2 Newton's law of cooling states that the rate of change of temperature in a cooling body is proportional to the difference in temperature between the body and its surroundings. Use t for time, x for temperature of the cooling body and xo for the temperature of the surroundings (assumed to be constant). 3 The rate at which a radioactive element changes from one form to another is k times the amount left unchanged, where k is a constant. Use t for time, A be the amount of unchanged substance and v be the amount which has been changed at time t.

4 Let the rate of production of a colony of bacteria be proportional to the number present and rate of destruction be proportional to the square of the number present. Write down a differential equation describing the rate of growth of the colony. Let N be the number of bacteria present.

2

5 The rate at which the height h of a certain plant increases is proportional to the natural logarithm of the difference between its present height, h, and its final height H. 6 The manufacturers of a certain brand of soap powder are concerned that the number, n, of people buying their product at any time t has remained constant for some months. They launch a major advertising programme which results in the number of customers increasing at a rate proportional to the square root of n. Express as differential equations the progress sales i) before advertising ii) after advertising. 7 Two quantities x and y each vary with time t. Express the following statements as differential equations involving x, y and t: i) the rate of change of y with respect to t is inversely proportional to x ii) the rate of change of x with respect to t is inversely proportional to y. Hence form a differential equation involving x and y only.

Applications of Differential Equations 1. A radioactive substance decays at a rate proportional to the amount present. If one gram of a radioactive substance reduces to

3

¼ gram in four hours, find how long it will be until

1 gram 10

remains. 2. A certain radioactive substance is known to decay at a rate proportional to the amount present. A block of this substance having a mass of 100 grams originally is observed. After 40 hours, its mass reduces to 90 grams. Find i) an expression for the mass of the substance at any time, ii) the time-lapse before the block decays to half of its original mass. 3. The rate of increase of the population of bacteria in a culture is proportional to the number of bacteria present at any given instant. Assume the initial count of bacteria is 1000 and after one hour the count is 1200. Find the number of bacteria present immediately after 5 hours ii) the time-lapse before the number reaches 4000. i)

4. Two quantities x and y each vary with time t. Express the following statements as differential equations involving x, y and t: i) the rate of change of y with respect to t is inversely proportional to x. ii) the rate of change of x with respect to t is inversely proportional to y. Hence form a differential equation involving x and y only, and prove that the solution is y = Ax n , where A and n are constants. 5.

Solve the differential equation when x = 0.

dy

dx = e

x + y , given that y = 0

6. Suppose a metal rod is immersed in water. The water temperature is held constant at 15 °C. It is observed that the rod cools from 60°C to 40°C in 3 minutes. Find by means of Newton's law of cooling i) an expression for the temperature of the rod at any time t, ii) the temperature of the rod immediately after 9 minutes.

7. The rate at which a radioactive element changes from one form to another is k times the amount left unchanged, where k is a constant. At time t = 0, the amount of unchanged substance is A and the amount which has been changed at time t is v. Write down a differential equation for v, and solve it to give v in terms of A, k and t. Sketch the graph of v against t. Find the half-life of the

4

element, i.e. the time needed for half of the amount A to be changed. 8. A cylindrical tank of radius 20 cm stands with its axis of symmetry vertical. Water is running out of the tank, through an open tap, at a rate which is proportional to the square root of the volume of the water remaining in the tank. (a) Express this information in the form of a differential equation. (b) Given that the depth of water in the tank is initially 38.5 cm and that the tank empties completely in 55 seconds, show that V = (220 − 4t ) 2 , where V is the volume in cm 3 and t is the time in seconds. (c) The tank is emptied completely but the person responsible forgot to close the tap. Water is added to the tank at a rate of 1320 cm 3 per second but the tap remains open. Find the depth to which the water rises in the tank. (Give your answer to 1 decimal place. Take π = 22/7). 9. A new and wonderful computer comes on the market. The rate of increase in the number of people, x, who owns one is proportional to the product of x and (N – x), where N is a constant. (a) Write down a differential equation relating x and t. 1 N , and after one week of aggressive selling, x = (b) Initially x = 10 1 x N . Show that after t weeks, = 3t − 2 . 4 N−x (c) What happens to x as t → ∞ ? (d) Sketch the graph of x against t. (e) Find the time at which ¾ N people own one of these computers. 10. The rate of destruction of a drug by the kidneys is proportional to the amount of the drug present in the body. The constant of proportionality is denoted by k. At time t the quantity of drug in the body is x. Write down a differential equation relating x and t, and show that the general solution is x = Ae − kt , where A is an arbitrary constant. Before t = 0 there is no drug in the body, but at t = 0, a quantity Q of the drug is administered. When t = 1 the amount of drug in the body is Qα, where α is a constant such that 0 < α < 1. Show that x = Qα t. Sketch the graph of x against t for 0 < t < 1. When t = 1 and again when t = 2 another dose Q is administered. Show that the amount of drug in the body immediately after t = 2 is Q(1 + α + α2). Answers: 1. t = 6.64 hours 2. i) y = 100e −0.00263t

ii) 263hrs

5

3. i) 2490

ii) 7.60 hour

4. i)

dy dt

=

a x

ii)

dx dt

=

b y

5. y = −ln(2 − e x )

t

6 i)

5 x = 15 + 45  3 or 9

x = 15 + 45e -0.196 t

ii) x = 22.716 = 22.7°C

7. v = A(1 - e -kt ) ; t = 1 k ln2 8. c) 21.7 cm 9. c) Ans: x → N e) 3 weeks More exercises 1. Newton's law of cooling states that the rate of decrease of temperature of a hot body is proportional to the excess of the temperature of the body over that of the surroundings. Using t for time in minutes, θ for temperature in degree C, and θo for the temperature of the surroundings (assumed constant), express the law in the form of a differential equation. In a particular case θ o = 20, θ = 80 when t = 0, t 5 and θ = 70 when t = 5. Prove that, at time t, and θ = 20 + 60 ( ) 5 6 find, to the nearest tenth of a minute, the value of t when θ = 30. 2. Waste from a mining operation is dumped on a 'slag-heap', which is a large mound, roughly conical in shape, which continually increases in size as more waste material is added to the top. In a mathematical model, the rate at which the height h of the slag-heap increases is inversely proportional to h 2 . Express this statement as a differential equation relating h with the time t. dh 1 ∝ dt h2

Show by integration that the general solution of the differential equation relating h and t may be expressed in the form h 3 = At + B , where A and B are constants. h3 = At + B A new slag-heap was started at time t = 0, and after 2 years its height was 18 m. Find the time by which its height would grow to 30 m. The assumptions underlying this mathematical model are that the volume V of the slag-heap increases at a constant rate, and that the slag-heap remains the same shape as it grows, so that V is proportional to h3. Show how these assumptions lead to the model described in the first paragraph. 3. A tank, with rectangular base and having vertical sides of height h, is initially full of water. The water leaks out of a small hole in the horizontal base of the tank at a rate which, at any instant, is proportional to the square root of the depth of the water at that instant. If x is the depth of the water at time t after the leak started, write down a differential equation connecting x and t. If the tank is exactly half empty after one hour, find the further time that elapses before the tank becomes completely empty. 6

4. A vessel contains liquid which is flowing out from a small hole at a point O in the base of the vessel. At time t, the height of the liquid surface above O is z and the rate at which z is decreasing is 3

inversely proportional to z 2 . Write down a differential equation which expresses dz dt in terms of z. Given that z = 25 when t = 0 and that z = 16 when t = 30, find the value of t when z = 0. 5. The positive quantities x and y are related by the differential 2 equation dy =  y  . dt x (a) Find the general solution of this differential equation, expressing y in terms of x. (b) Show that the number of the family of solution curves which passes through the point (1, 1) is a straight line, and that the member of the family which passes through (1, 2) has an asymptote parallel to the y-axis.

6. Under suitable conditions, the rate at which the temperature of a hot object decreases may be taken to be proportional to the difference between the temperature of the hot object and the temperature of its surroundings. Express this information as a differential equation connecting θ and t, whereθ is the temperature of the hot object at time t, for the case where the temperature of the surroundings has the constant (non-zero) value θo. Show that the general solution of the differential equation may be expressed in the form θ = θ o + A e- kt, where A and k are constants. Given that θ = 5θ o when t = 0 and that θ = 3θ o when t = T, find, in terms of T, the value of t when θ = 2θ o . dx = kx , dt where k is a positive constant, expressing x in terms of k and t in your answer. dx = kx(a - x), 0 < x < a, where k (b) Solve the differential equation dt and a are positive constants, given that x = ½ a when t = 0. Express x in terms of k, a and t in your answer.

7. (a) Find the general solution of the differential equation

8. The hot water in a domestic hot-water tank is at temperature θ°C at time t minutes. A dripping hot-water tap means that water at temperature θ°C is being taken from the tank at a steady rate and being replaced by cold water, at temperature 10°C, at the same rate. It is assumed that the incoming cold water mixes immediately with the hot water already in the tank, that the tank is

7

fully insulated, and that no heat is being supplied by a waterheater. Under these circumstances the rate at which the temperature of the water in the tank drops is given by the dθ = − k(θ − 10) , where k is a positive constant differential equation dt whose numerical value depends on the rate at which the tap is dripping. At time t = 0 the water in the tank is at temperature 60° C. (a) Solve the differential equation, expressing θ in terms of k and t in your answer. (b) When heat is being supplied by a water-heater, the differential dθ = − k(θ − 10) + h , where equation relating θ and t becomes dt the numerical value of the constant h depends on the rate at which heat is being supplied. Describe, with the aid of a diagram, what will happen to the temperature of the water in the tank for each of the cases h = 40k and h = 60k, again assuming that θ = 60 when t = 0.

9. At time t, the volume of the water in a storage tank is denoted by v. Water is lost from the tank by leakage at a rate equal to kv where k is a positive constant. Water is also lost through evaporation at a constant rate p. a) Write down a differential equation expressing this information. − kt − p (1 − e − kt ) b) Given that v = V when t = 0, show that v = Ve k c) Find the time, t, taken for the tank to empty. 10.In a model of mortgage repayment, the sum of money owed to the Building society is denoted by x and the time is denoted by t. Both x and t are taken to be continuous variables. The sum of money owed to the Building society increases, due to interest, at a rate proportional to the sum of money owed. Money is also repaid at a constant rate r. When x = a, interest and repayment balance. dx r =  (x - a) . Given that, when t = 0, x = A, Show that for x > 0, dt a find x in terms of t, r, a and A. On a single, clearly labeled sketch, show the graph of x against t in the two cases (i) A > a, (ii) A < a. State the circumstances under which the loan is repaid in a finite a  a  time T and show that, in this case T = ln . r a - A

8

dx dt

dx

∝ x and

x=a⇒ Thus

dx dt

dt

dx dt =

r a

∝ −r ⇒

= 0. ∴ k =

dx dt

= kx − r

r a

(x − a)

Integrate : ln (x - a) =

r a

rt t ⇒ x = a + (A − a)e

a

x A
x=a

Aa, the loan is repaid, i.e x = 0  x - a  = ln - a  = rT ⇒ T = a ln a       r  A-a  A-a  A-a a

when x = 0, ln Thus

dx dt

=

r a

(x − a)

Integrate : ln (x - a) =

r a

rt t ⇒ x = a + (A − a)e

a

11.A population of insects is allowed to grow in an experimental environment. The rate of increase of the population is proportional to the number, n, of insects, at any time t days after the start of the experiment. Regarding n and t as continuous variables, form a differential equation relating n and t and solve it to show that n = Ae kt , where A and k are constants. The net increases during the fourth and fifth days are 350 and 500 insects respectively. Determine the population at the beginning of the fourth day. Hence, or otherwise, determine the population at the beginning of the first day. 12.A race called the Matrices live on an isolated island called Vector. The number of births per unit time is proportional to the population at any time and the number of deaths per unit time is

9

proportional to the square of the population. If the population at dp = ap - bp 2 , where a and b are positive time t is p, show that dt constants. Solve the equation for p in terms of t, given that 2 a p = ( ) when t = 0. Show that there is a limit to the size of the 3 b population. 13.Newton's law of cooling states that the rate of loss of temperature of a cooling body is proportional to the difference in temperature between the body and its surroundings. Express this law in the form of a differential equation, using symbols t for time and x for the temperature of the body. Solve this equation for x in terms of t, given that the room temperature is 20°C and that it takes a particular body 12 minutes to cool from 100°C to 50°C. Hence find the time taken by the body to cool from 50°C to 25°C. −

dx ∝ (x - x0 ) ; x - x0 = Ae -kt ; 21.9 mins dt

14.Newton's law of cooling states that the rate of loss of temperature of a cooling body is proportional to the difference between the body temperature and the surrounding temperature. i) Set up a differential equation describing the temperature of the body. −

dθ dt

= k(θ − θ o )

ii) A beaker of water initially at 100°C is allowed to cool in a room maintained at 20°C. After 2 minutes, the water temperature is 80°C. Find the as a function of time. t θ = 20 + 80 ( 3 4 ) 2

or θ = 20 + 80 e - 0.144t

iii) Evaluate (a) the water temperature after 4 minutes, θ = 20 + 80 ( 3 65 ° C

(b) the time taken for the water to reach 50°C.

4

4 ) 2 =

t = 6.82 mins.

Answer: 1. when θ = 30 , t = 49.1 mins.

2. 9.3 years

dx 3. − dt = k x ;

2 +1

10

dz 1 4. − dt ∝ 3 ; 44.6 2 z

9. −

dv dt

= kv − p

11