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dx
Step 3. Using the deп¬Ѓnition of the indeп¬Ѓnite integral, we antidiп¬Ђerentiate:

e4x y = (10e4x )dx + c.

Step 4. We multiply both sides by eв€’4x .

y = eв€’4x (10e4x )dx + c

e4x
в€’4x
10 В·
y=e +c
4
10
+ ceв€’4x .
y(x) = (8)
4
4.3. LINEAR FIRST ORDER DIFFERENTIAL EQUATIONS 167

Step 5. We impose the condition y(0) = 200 to solve for c.
5 5
c = 200 в€’ .
y(0) = 200 = + c,
2 2
Step 6. We replace c by its value in solution (8)

5 5
eв€’4x .
+ 200 в€’
y(x) =
2 2

Exercises 4.3 Find y(t) in each of the following:

2. y = в€’2y, y(0) = 1200
1. y = 4y, y(0) = 100

dy
y = в€’4(y в€’ c), y(0) = y0
3. 4. L + Ry = E, y(0) = y0
dt

5. y + 3y(t) = 32, y(0) = 0 6. y = ty, y(0) = y0

t2
1 dy 1
Hint: y = ty, dt = tdt; dy = + c.
y dt y 2

7. The population P (t) of a certain country is given by the equation:

P (t) = 0.02P (t), P (0) = 2 million.

(i) Find the time when the population will double.
(ii) Find the time when the population will be 3 million.

8. Money grows at the rate of r% compounded continuously if
r
A (t) = A(t), A(0) = A0 ,
100
where A(t) is the amount of money at time t.

(i) Determine the time when the money will double.
(ii) If A = \$5000, determine the time for which A(t) = \$15,000.
168 CHAPTER 4. APPLICATIONS OF DIFFERENTIATION

9. A radioactive substance satisп¬Ѓes the equation

A (t) = в€’0.002A(t), A(0) = A0 ,

where t is measured in years.
1
(i) Determine the time when A(t) = A0 . This time is called the
2
half-life of the substance.
(ii) If A0 = 20 grams, п¬Ѓnd the time t for which A(t) equals 5 grams.

10. The number of bacteria in a test culture increases according to the equa-
tion
N (t) = rN (t), n(0) = N0 ,
where t is measured in hours. Determine the doubling period. If N0 =
100, r = 0.01, п¬Ѓnd t such that N (t) = 300.

11. NewtonвЂ™s law of cooling states that the time rate of change of the tem-
perature T (t) of a body is proportional to the diп¬Ђerence between T and
the temperature A of the surrounding medium. Suppose that K stands
for the constant of proportionality. Then this law may be expressed as

T (t) = K(A в€’ T (t)).

Solve for T (t) in terms of time t and T0 = T (0).

12. In a draining cylindrical tank, the level y of the water in the tank drops
according to TorricelliвЂ™s law

y (t) = в€’Ky 1/2

for some constant K. Solve for y in terms of t and K.

13. The rate of change P (t) of a population P (t) is proportional to the
square root of P (t). Solve for P (t).

14. The rate of change v (t) of the velocity v(t) of a coasting car is propor-
tional to the square of v. Solve for v(t).

In exercises 15вЂ“30, solve for y.
4.4. LINEAR SECOND ORDER HOMOGENEOUS DIFFERENTIAL EQUATIONS169

16. y + 3x2 y = 0, y(0) = 6
15. y = x в€’ y, y(0) = 5

17. xy (x) + 3y(x) = 2x5 , y(2) = 1 18. xy + y = 3x2 , y(1) = 4

19. y + y = ex , y(0) = 100 20. y = в€’6xy, y(0) = 9

22. y = xy 3 , y(0) = 2
21. y = (sin x)y, y(0) = 5
в€љ
1+ x
24. y в€’ 2y = 1, y(1) = 3
23. y = в€љ , y(0) = 10
1+ y

25. y = ry в€’ c, y(0) = A 26. y в€’ 3y = 2 sin x, y(0) = 12
3
27. y в€’ 2y = 4e2x , y(0) = 4 28. y в€’ 3x2 y = ex , y(0) = 7

1
30. y в€’ 3y = e2x , y(0) = 1
29. y в€’ y = sin x, y(1) = 3
2x

4.4 Linear Second Order Homogeneous Dif-
ferential Equations
Deп¬Ѓnition 4.4.1 A linear second order diп¬Ђerential equation in the variable
y is an equation of the form

y + p(x)y + q(x)y = r(x).

If r(x) = 0, we say that the equation is homogeneous; otherwise it is called
non-homogeneous. If p(x) and q(x) are constants, we say that the equation
has constant coeп¬ѓcients.

Deп¬Ѓnition 4.4.2 If f and g are diп¬Ђerentiable functions, then the Wronskian
of f and g is denoted W (f, g) and deп¬Ѓned by

W (f, g) = f (x)g (x) в€’ f (x)g(x).

Example 4.4.1 Compute the following Wronskians:
170 CHAPTER 4. APPLICATIONS OF DIFFERENTIATION

(ii) W (epx sin(qx), epx cos(qx))
(i) W (sin(mx), cos(mx))

(iii) W (xn , xm ) (iv) W (x sin(mx), x cos(mx))

d d
(cos(mx)) в€’
Part (i) W (sin mx, cos mx) = sin(mx) (sin(mx)) cos(mx)
dx dx
= в€’m sin2 (mx) в€’ m cos2 (mx)
= в€’m(sin2 (mx) + cos2 (mx))
= в€’m

Part (iii) W (epx sin qx, epx cos qx)
= epx sin qx(pepx cos qx в€’ qepx sin qx)
в€’epx cos qx(pepx sin qx + qepx cos qx)
= в€’qe2px (sin2 qx + cos2 qx)
= в€’qe2px .

Part (iii) W (xn , xm ) = xn В· mxmв€’1 в€’ xm В· nxnв€’1
= (m в€’ n)xn+mв€’1 .

Part (iv) W (x sin mx, x cos mx) = (x sin mx)(cos mx в€’ mx sin mx)
в€’(x cos mx)(sin mx + mx cos mx)
= в€’mx2 (sin2 mx + cos2 mx)
= в€’mx2 .

Deп¬Ѓnition 4.4.3 Two diп¬Ђerentiable functions f and g are said to be linearly
independent if their Wronskian, W (f (x), g(x)), is not zero for all x in the
domains of both f and g.

Example 4.4.2 Which pairs of functions in Example 8 are linearly indepen-
dent?
(i) In Part (i), W (sin mx, cos mx) = в€’m = 0 unless m = 0. Therefore,
sin mx and cos mx are linearly independent if m = 0.
(ii) In Part (ii),
W (epx sin qx, epx cos qx) = в€’qe2px в‰Ў if q = 0.
Therefore, epx sin(qx) and epx cos qx are linearly independent if q = 0.
4.4. LINEAR SECOND ORDER HOMOGENEOUS DIFFERENTIAL EQUATIONS171

(iii) In Part (iii), W (xn , xm ) = (m в€’ n)xn+mв€’1 в‰Ў 0 if m = n. Therefore, if m
and n are not equal, then xn and xm are linearly independent.

(iv) In Part (iv),
W (x sin mx, x cos mx) = в€’mx2 в‰Ў 0
if m = 0. Therefore, x sin mx and x cos mx are linearly independent if
m = 0.

Theorem 4.4.1 Consider the linear homogeneous second order diп¬Ђerential
equation
y + p(x)y + q(x)y = 0. (1)

(i) If y1 (x) and y2 (x) are any two solutions of (1), then every linear combi-
nation y(x), with constants A and B,

y(x) = Ay1 (x) + By2 (x)

is also a solution of (1).

(ii) If y1 (x) and y2 (x) are any two linearly independent solutions of (1), then
every solution y(x) of (1) has the form

y(x) = Ay1 (x) + By2 (x)

for some constants A and B.

Proof.
Part (i) Suppose that y1 and y2 are solutions of (1), A and B are any con-
stants. Then

(Ay1 + By2 ) + p(Ay1 + By2 ) + q(Ay1 + By2 )
= Ay1 + By2 + Apy1 + ABy2 + Aqy1 + BqBy2
= A(y1 + py1 + qy1 ) + B(y2 + py2 + qy2 )
= A(0) + B(0) (Because y1 and y2 are solutions of (1))
= 0.

Hence, y = Ay1 + By2 are solutions of (1) whenever y1 and y2 are solutions
of (1).
172 CHAPTER 4. APPLICATIONS OF DIFFERENTIATION

Part (ii) Let y be any solution of (1) and suppose that

y = Ay1 + By2 (2)
(3)
y = Ay1 + By2

We solve for A and B from equations (2) and (3) to get

yy2 в€’ y2 y W (y, y2 )
A= =
y1 y2 в€’ y2 y1 W (y1 , y2 )
y1 y в€’ y1 y W (y1 , y)
B= = .
y1 y2 в€’ y2 y1 (y1 , y2 )

Since y1 and y2 are linearly independent, W (y1 , y2 ) = 0, and hence, A and B
are uniquely determined.

Remark 15 It turns out that the Wronskian of two solutions of (1) is either
identically zero or never zero for any value of x.

Theorem 4.4.2 Let y1 and y2 be any two solutions of the homogeneous equa-
tion
y + py + qy = 0. (1)

Let
W (x) = W (y1 , y2 ) = y1 (x)y2 (x) в€’ y1 (x)y2 (x).
Then

W (x) = в€’pW (x)
W (x) = ceв€’ p(x)dx

for some constant c. If c = 0, then W (x) = 0 for every x. If c = 0, then
W (x) = 0 for every x.
Proof. Since y1 and y2 are solutions of (1),

y1 = в€’py1 в€’ qy1 (2)
y2 = в€’py2 в€’ qy2 (3)
4.4. LINEAR SECOND ORDER HOMOGENEOUS DIFFERENTIAL EQUATIONS173

Then,

W (x) = (y1 y2 в€’ y1 y2 )
= y1 y2 + y1 y2 в€’ y1 y2 в€’ y1 y2
= y1 y2 в€’ y2 y1
= y1 (в€’py2 в€’ qy2 ) в€’ y2 (в€’py1 в€’ qy1 ) (from (2) and (3))
= в€’p[y1 y2 в€’ y2 y1 ]
= в€’pW (x).

Thus,
W (x) + pW (x) = 0.
By Theorem 4.3.1

0 dx + c = ceв€’
W (x) = eв€’ pdx
pdx
.

If c = 0, W (x) в‰Ў 0; otherwise W (x) is never zero.

Theorem 4.4.3 (Homogeneous Second Order) Consider the linear second
order homogeneous diп¬Ђerential equation with constant coeп¬ѓcients:

ay + by + cy = 0, a = 0. (1)

(i) If y = emx is a solution of (1), then

am2 + bm + c = 0. (2)

Equation (2) is called the characteristic equation of (1).
в€љ в€љ
в€’b в€’ b2 в€’ 4ac в€’b + b2 в€’ 4ac
(ii) Let m1 = and m2 = . Then the follow-
2a 2a
ing three cases arise:

Case 1. The discriminant b2 в€’ 4ac > 0. Then m1 and m2 are real and
distinct. The two linearly independent solutions of (1) are em1 x and
em2 x and its general solution has the form

y(x) = Aem1 x + Bem2 x .
174 CHAPTER 4. APPLICATIONS OF DIFFERENTIATION

Case 2. The discriminant b2 в€’ 4ac = 0. Then m1 = m2 = m, and only
one real solution exists for equation (2). The roots are repeated. In
this case, emx and xemx are two linearly independent solutions of (1)
and the general solution of (1) has the form

y(x) = Aemx + Bxemx = emx (A + Bx).

Case 3 b2 в€’ 4ac < 0. Then m1 = p в€’ iq, and m2 = p + iq where p =
в€љ
в€’b/2a, and q = 4ac в€’ b2 /2a. In this case, the functions epx sin qx
and epx cos qx are two linearly independent solutions of (1) and the
most general solution of (1) has the form

y(x) = epx (A sin qx + B sin qx).

Proof. Let y = emx . Then y = memx , y = m2 emx and

ay + by + cy = (am2 + bm + c)emx = 0, a = 0 в†”
am2 + bm + c = 0, a = 0 в†”
в€љ
b2 в€’ 4ac
в€’b
В± .
m=
2a 2a
This proves Part (i).

Case 1. For Case 1, em1 x and em2 x are solutions of (1). We show that these
are linearly independent by showing that their Wronskian is not zero.

W (em1 x , em2 x ) = em1 x В· m2 em2 x в€’ m1 em1 x В· em2 x
= (m2 в€’ m1 )e(m1 +m2 )x .

Since m1 = m2 , W (em1 x , em2 x ) = 0.

Case 2. We already know that emx = 0, and m = в€’b/2a. Let us try
y = xemx . Then

ay + by + cy = a(2m + m2 x)emx + b(1 + mx)emx + xemx
= (b + 2am)emx + (am2 + bm + c)xemx
= (b + 2a(в€’b/2a))emx
= 0.
4.4. LINEAR SECOND ORDER HOMOGENEOUS DIFFERENTIAL EQUATIONS175

Therefore, emx and xemx are both solutions. We only need to show that
they are linearly independent.
W (emx , xemx ) = emx (emx + mxemx ) в€’ memx (xemx )
= e2mx + mxe2mx в€’ mxe2mx
= e2mx
= 0.
Hence, emx and xemx are linearly independent and the general solution
of (1) has the form
y(x) = Aemx + Bxemx = emx (A + Bx).

Case 3. In Example 8, we showed that
W (epx sin qx, epx cos qx) = в€’qe2px = 0
since q = 0.
We only need to show that epx sin qx and epx cos qx are solutions of (1).
Let y1 = epx sin qx and y2 = epx cos qx. Then,
y1 = pepx sin qx + qepx cos qx
y1 = p2 epx sin qx + pqepx cos qx + pqepx cos qx в€’ q 2 sin qx

ay1 + by1 + cy1 = aepx (p2 sin qx + 2pq cos qx в€’ q 2 sin qx)
+ bepx (p sin qx + q cos qx) + cepx sin qx
= epx sin qx[a(p2 в€’ q 2 ) + (bp + c)] + epx cos qx[2apq + bq]
b2 4ac в€’ b2 в€’b
px
в€’
= e sin(qx) a +c
+b
4a2 4a2 2a
в€љ
в€’b
px
4ac в€’ b2
+ e cos qx 2a +b
2a
b2 в€’ 2ac + b2 + 2ac
px
= e sin(qx)
2a
+ epx cos(qx)
= 0.
Therefore y1 = epx sin(qx) is a solution of (1). Similarly, we can show
that y2 = epx cos qx is a solution of (1). We leave this as an exercise.
176 CHAPTER 4. APPLICATIONS OF DIFFERENTIATION

Remark 16 Use Integral Tables or computer algebra in evaluating the in-
deп¬Ѓnite integrals as needed.

Example 4.4.3 Solve the diп¬Ђerential equations for y(t).
(i) y в€’ 5y + 14y = 0
(ii) y в€’ 6y + 9y = 0
(iii) y в€’ 4y + 5y = 0
We let y = emt . We then solve for m and determine the solution. We observe
that y = memt , y = m2 emt .
Part (i) By substituting y = emt in the equation we get
m2 emt в€’ 5memt + 14emt = emt (m2 в€’ 5m + 14) = 0 в†’
m2 в€’ 5m + 14 = 0 = (m в€’ 7)(m + 2) в†’ m = 7, в€’2.
Therefore,
y(t) = Aeв€’2t + Be7t .

Part (ii) Again, by substituting y = emt , we get
m2 emt в€’ 6memt + 9emt = 0
m2 в€’ 6m + 9 = (m в€’ 3)2 = 0
m = 3, 3.
The solution is
y(t) = Ae3t + Bte3t .

Part (iii) By substituting y = emt , we get
m2 emt в€’ 4memt + 5emt = emt (m2 в€’ 4m + 5) = 0 в†’
m2 в€’ 4m + 5 = 0
в€љ
4 В± 16 в€’ 20
= 2 В± 1i.
m=
2
The general solution is
y(t) = e2t (A cos t + B sin t).
4.4. LINEAR SECOND ORDER HOMOGENEOUS DIFFERENTIAL EQUATIONS177

Example 4.4.4 Solve the diп¬Ђerential equation for y(t) where y(t) satisп¬Ѓes
the conditions

y (t) в€’ 2y (t) в€’ 15y(t) = 0; y(0) = 1, y (0) = в€’1.

We assume that y(t) = emt . By substitution we get the characteristic equa-
tion:
m2 в€’ 2m в€’ 15 = 0, m = 5, в€’3.
The general solution is
y(t) = Aeв€’3t + Be5t .
We now impose the additional conditions y(0) = 1, y (0) = в€’1.

y(t) = Aeв€’3t + Be5t
y (t) = в€’3Aeв€’3t + 5Be5t
y(0) = A + B = 1
y (0) = в€’3A + 5B = в€’1

On solving these two equations simultaneously, we get
3 1
A= , B= .
4 4
Then the exact solution is
3 в€’3t 1 5t
y(t) = e + e.
4 4

Exercises 4.4 Solve for y(t) from each of the following:

1. y в€’ y в€’ 20y = 0

2. y в€’ 8y + 16y = 0

3. y + 9y + 20y = 0

4. y + 4y + 4 = 0

5. y в€’ 8y + 12y = 0

6. y в€’ 6y + 10y = 0
178 CHAPTER 4. APPLICATIONS OF DIFFERENTIATION

7. y в€’ y в€’ 6y = 0, y(0) = 10, y (0) = 15
8. y в€’ 4y + 4y в€’ 0, y(0) = 4, y (0) = 8
9. y + 8y + 12y = 0, y(0) = 1, y (0) = 3
10. y + 6y + 10y = 0, y(0) = 5, y (0) = 7
11. y в€’ 4y = 0, y(0) = 1, y (0) = в€’1
12. y в€’ 9y = 0, y(0) = в€’1, y (0) = 1
13. y + 9y = 0, y(0) = 2, y (0) = 3
14. y + 4y = 0, y(0) = в€’1, y (0) = 2
15. y в€’ 3y + 2y = 0, y(0) = 2, y (0) = в€’2
16. y в€’ y в€’ 6y = 0, y(0) = 6, y (0) = 5
17. y + 4y + 4y = 0, y(0) = 1, y (0) = 4
18. y в€’ 6y + 9y = 0, y(0) = 1, y (0) = в€’1
19. y + 6y + 13y = 0, y(0) = 1, y (0) = 2
20. y в€’ 3y + 2y = 0
21. y + 3y + 2y = 0
22. y + m2 y = 0
23. y в€’ m2 y = 0
24. y + 2my + m2 y 2 = 0
25. y + 2my + (m2 + 1)y = 0
26. y в€’ 2my + (m2 + 1)y = 0
27. y + 2my + (m2 в€’ 1)y = 0
28. y в€’ 2my + (m2 в€’ 1)y = 0
29. 9y в€’ 12y + 4y = 0
30. 4y + 4y + y = 0
4.5. LINEAR NON-HOMOGENEOUS SECOND ORDER DIFFERENTIAL EQUATIONS179

4.5 Linear Non-Homogeneous Second Order
Diп¬Ђerential Equations
Theorem 4.5.1 (Variation of Parameters) Consider the equations
y + p(x)y + q(x)y = r(x) (1)
y + p(x)y + q(x)y = 0. (2)
Suppose that y1 and y2 are any two linearly independent solutions of (2).
Then the general solution of (1) is
y1 (x)r(x) y2 (x)r(x)
dx в€’ y1 (x)
y(x) = c1 y1 (x) + c2 y2 (x) + y2 (x) dx .
W (y1 , y2 ) W (y1 , y2 )
Proof. It is already shown that c1 y1 (x) + c2 y2 (x) is the most general solution
of the homogeneous equation (2), where c1 and c2 are arbitrary constants.
We observe that the diп¬Ђerence of any two solutions of (1) is a solution of (2).
Suppose that y в€— (x) is any solution of (1). We wish to п¬Ѓnd two functions,
u1 and u2 , such that
y в€— (x) = u1 (x)y1 (x) + u2 (x)y2 (x). (3)
By diп¬Ђerentiation of (3), we get
y в€— (x) = (u1 y1 + u2 y2 ) + (u1 y1 + u2 y2 ). (4)
We impose the following condition (5) on u1 and u2 :
u1 y1 + u2 y2 = 0. (5)
Then
y в€— (x) = u1 y1 + u2 y2
y в€— (x) = (u1 y1 + u2 y2 ) + u1 y1 + u2 y2 .
Since y в€— (x) is a solution of (1), we get
r(x) = y в€— + p(x)y в€— + q(x)y в€—
= (u1 y1 + u2 y2 ) + (u1 y1 + u2 y2 ) + p(x)[u1 y1 + u2 y2 ]
+ q(x)(u1 y1 + u2 y2 )
= u1 [y1 + p(x)y1 + q(x)y1 ] + u2 [y2 + p(x)y2 + q(x)y2 ]
+ (u1 y1 + u2 y2 )
= u1 y1 + u2 y2 .
180 CHAPTER 4. APPLICATIONS OF DIFFERENTIATION

Hence, another condition on u1 and u2 is
u1 y1 + u2 y2 = r(x). (6)
By solving equations (5) and (6) simultaneously for u1 and u2 , we get
в€’y2 r(x) y1 r(x)
u1 = and u2 (x) = . (7)
y1 y2 в€’ y2 y1 y1 y2 в€’ y2 y1
The denominator of the solution (7) is the Wronskian of y1 and y2 , which is
not zero for any x since y1 and y2 are linearly independent by assumption.
By taking the indeп¬Ѓnite integrals in equation (7), we obtain u1 and u2 .
y1 (x)r(x)
y2 (x)r(x)
u1 (x) = в€’ dx and u2 (x) = dx.
W (y1 , y2 ) W (y1 , y2 )
By substituting these values in (3), we get a particular solution
y в€— (x) = y2 u2 + y1 u1
y1 (x)r(x) y2 (x)r(x)
dx в€’ y1 (x)
= y2 (x) dx.
W (y1 , y2 ) W (y1 , y2 )
This solution y в€— (x) is called a particular solution of (1). To get the general
solution of (1), we add the general solution c1 y1 (x) + c2 y2 (x) of (2) to the
particular solution of y в€— (x) and get

y1 (x)r(x) y2 (x)r(x)
dx в€’ y1 (x)
y(x) = (c1 y1 (x) + c2 y2 (x)) + y2 (x) dx .
W (y1 , y2 ) W (y1 , y2 )

This completes the proof of this theorem.

Remark 17 The general solution of (2) is called the complementary solution
of (1) and is denoted yc (x).
yc (x) = c1 y1 (x) + c2 y2 (x).
The particular solution y в€— of (1) is generally written as yp .
r(x)y2 (x)
r(x)y1 (x)
dx в€’ y1 dx.
yp = y2
W (y1 , y2 ) W (y1 , y2 )
The general solution y(x) of (1) is the sum of yc and yp ,
y = yc (x) + yp (x).
4.5. LINEAR NON-HOMOGENEOUS SECOND ORDER DIFFERENTIAL EQUATIONS181

Example 4.5.1 Solve the diп¬Ђerential equation

y + 8y + 12y = eв€’3x .

We п¬Ѓnd the general solution of the homogeneous equation

y + 8y + 12y = 0.

We let y = emx be a solution. Then y = memx , y = m2 emx and

m2 emx + 8memx + 12emx = 0
m2 + 8m + 12 = 0 m = в€’6, в€’2.

So,
yc (x) = Aeв€’6x + Beв€’2x
is the complementary solution. We compute the Wronskian

W (eв€’6x , eв€’2x ) = eв€’6x (в€’2)eв€’2x в€’ eв€’2x (в€’6)eв€’6x
= eв€’8x (в€’2 + 6)
= 4eв€’8x
= 0.

By Theorem 4.4.1, the particular solution is given by
eв€’6x В· eв€’3x eв€’2x В· eв€’3x
в€’2x в€’6x
dx в€’ e
yp = e dx
4eв€’8x 4eв€’8x
1 в€’x 1 3x
= eв€’2x e dx в€’ eв€’6x e dx
4 4
1 в€’x 1 3x
= в€’eв€’2x в€’ eв€’6x
e e
4 4
1 1 в€’3x
= в€’ eв€’3x в€’ e
4 12
1
= в€’ eв€’3x .
3
The complete solution is the sum of the complementary solution yc and the
particular solution yp .
1 в€’3x
y(t) = Aeв€’6x + Beв€’2x в€’ e.
3
182 CHAPTER 4. APPLICATIONS OF DIFFERENTIATION

Exercises 4.5 Find the complementary, particular and the complete solu-
tion for each of the following. Use tables of integrals or computer algebra to
do the integrations, if necessary.

2. y в€’ 9y = e2x
1. y + 4y = sin(3x)

4. y в€’ 4y = eв€’x
3. y + 9y = cos 2x

5. y в€’ y = xex 6. y в€’ 5y + 6y = 3e4x

7. y в€’ 4y + 4y = eв€’x 8. y + 5y + 4y = 2ex

10. y + 5y + 6y = x2 e2x
9. my в€’ py = mg

In exercises 11вЂ“20, compute the complete solution for y.

11. y + y = 4x, y(0) = 2, y (0) = 1

12. y в€’ 9y = ex , y(0) = 1, y (0) = 5

13. y в€’ 2y в€’ 3y = 4, y(0) = 2, y (0) = в€’1

14. y в€’ 3y + 2y = 4x

15. y + 4y = sin 2x

16. y в€’ 4y = e2x

17. y в€’ 4y = eв€’2x

18. y + 4y = cos 2x

19. y + 9y = 2 sin 3x + 4 cos 3x

20. y + 4y + 5y = sin x в€’ 2 cos x
Chapter 5

The Deп¬Ѓnite Integral

5.1 Area Approximation
In Chapter 4, we have seen the role played by the indeп¬Ѓnite integral in п¬Ѓnd-
ing antiderivatives and in solving п¬Ѓrst order and second order diп¬Ђerential
equations. The deп¬Ѓnite integral is very closely related to the indeп¬Ѓnite inte-
gral. We begin the discussion with п¬Ѓnding areas under the graphs of positive
functions.

Example 5.1.1 Find the area bounded by the graph of the function y =
4, y = 0, x = 0, x = 3.

graph

From geometry, we know that the area is the height 4 times the width 3 of
the rectangle.

Area = 12.

Example 5.1.2 Find the area bounded by the graphs of y = 4x, y = 0, x =
0, x = 3.

183
184 CHAPTER 5. THE DEFINITE INTEGRAL

graph

1
From geometry, the area of the triangle is times the base, 3, times the
2
height, 12.
Area = 18.

Example 5.1.3 Find the area bounded by the graphs of y = 2x, y = 0, x =
1, x = 4.

graph

1
The required area is covered by a trapezoid. The area of a trapezoid is
2
times the sum of the parallel sides times the distance between the parallel
sides.
1
Area = (2 + 8)(3) = 15.
2
в€љ
4 в€’ x2 , y =
Example 5.1.4 Find the area bounded by the curves y =
0, x = в€’2, x = 2.

graph

By inspection, we recognize that this is the area bounded by the upper half
of the circle with center at (0, 0) and radius 2. Its equation is
в€љ
2 2
x + y = 4 or y = 4 в€’ x2 , в€’2 в‰¤ x в‰¤ 2.
Again from geometry, we know that the area of a circle with radius 2 is
ПЂr2 = 4ПЂ. The upper half of the circle will have one half of the total area.
Therefore, the required area is 2ПЂ.
5.1. AREA APPROXIMATION 185

Example 5.1.5 Approximate the area bounded by y = x2 , y = 0, x = 0,
and x = 3. Given that the exact area is 9, compute the error of your
approximation.
Method 1. We divide the interval [0, 3] into six equal subdivisions at the
35
1
points 0, , 1, , and 2. Such a subdivision is called a partition of [0, 3].
2 22
We draw vertical segments joining these points of division to the curve. On
each subinterval [x1 , x2 ], the minimum value of the function x2 is at x2 .
1
2
The maximum value x2 of the function is at the right hand end point x2 .
Therefore,

graph

The lower approximation, denoted L, is given by
2 2
1 1 3 1 1 5 1
2
L = 0 В· + 12 В· + В· + (2)2 В· + В·
2 2 2 2 2 2 2
1 9 25
= В· 0+1+ +4+
2 4 4
27
в‰€ 8 В· 75.
=
4
This approximation is called the left-hand approximation of the area. The
error of approximation is в€’0.25.
The Upper approximation, denoted U , is given by
2 2 2
1 1 3 1 1 5 1 1
1
В· + 12 В· + В· + (2)2 В· + + (3)2 В·
В·
U=
2 2 2 2 2 2 2 2 2
1 1 9 25
= +1+ +4+ +9
2 4 4 4
1 91
=
2 4
91
в‰€ 11 В· 38.
=
8
186 CHAPTER 5. THE DEFINITE INTEGRAL

The error of approximation is +2.28.
This approximation is called the right-hand approximation.

Method 2. (Trapezoidal Rule) In this method, for each subinterval [x1 , x2 ],
we join the point (x1 , x2 ) with the point (x2 , x2 ) by a straight line and п¬Ѓnd
2
1
1
the area under this line to be a trapezoid with area (x2 в€’ x1 )(x2 + x2 ). We
2
1
2
add up these areas as the Trapezoidal Rule approximation, T , that is given
by

2 2
1 1 1
1 1 1
2 2
в€’0 1в€’
T= 0+ 1+
+
2 2 2 2 2 2
2 2
1 3
1 3 3 3 2
2
в€’1 2в€’
+ 2+
+ +1
2 2 2 2 2 2
2 2
1 5
1 5 5 5
32 +
+ 22
в€’2 3в€’
+
+
2 2 2 2 2 2
2 2 2
12 1 3 5
2 2
+ 32
0 +2В· + 2(1 ) + 2 В· + 2(2) + 2 В·
=
4 2 2 2
1 9 25
= 1+2+ +8+ +9
4 2 2
37
= 9 В· 25.
=
4

The error of this Trapezoidal approximation is +0.25.

Method 3. (SimpsonвЂ™s Rule) In this case we take two intervals, say [x1 , x2 ]в€Є
[x2 , x3 ], and approximate the area over this interval by

1
[f (x1 ) + 4f (x2 ) + f (x3 )] В· (x3 в€’ x1 )
6

1
and then add them up. In our case, let x0 = 0, x1 = , x2 = 1, x3 =
2
3 5
, x4 = 2, x5 = and x6 = 3. Then the SimpsonвЂ™s rule approximation, S,
2 2
5.1. AREA APPROXIMATION 187

is given by
2 2
12 1
1 3
2
(1)2 + 4 В· + 22 (1)
В· (1) +
0 +4В·
S= + (1)
6 2 6 2
2
12 5
+ 32 В· (1)
2 +4В·
+
6 2
2 2 2
12 1 3 5
2 2
+ 32
+2В·1 +4В· +2В·2 +4В·
= 0 +4
6 2 2 2
54
= = 9 = Exact Value!
6
For positive functions, y = f (x), deп¬Ѓned over a closed and bounded interval
[a, b], we deп¬Ѓne the following methods for approximating the area A, bounded
by the curves y = f (x), y = 0, x = a and x = b. We begin with a common
equally-spaced partition,

P = {a = x0 < x1 < x2 < x3 < . . . < xn = b},
bв€’a
i, for i = 0, 1, 2, . . . , n.
such that xi = a +
n

Deп¬Ѓnition 5.1.1 (Left-hand Rule) The left-hand rule approximation for A,
denoted L, is deп¬Ѓned by
bв€’a
В· [f (x0 ) + f (x1 ) + f (x2 ) + В· В· В· + f (xnв€’1 )].
L=
n

Deп¬Ѓnition 5.1.2 (Right-hand Rule) The right-hand rule approximation for
A, denoted R, is deп¬Ѓned by
bв€’a
В· [f (x1 ) + f (x2 ) + f (x3 ) + В· В· В· + f (xn )].
R=
n

Deп¬Ѓnition 5.1.3 (Mid-point Rule) The mid-point rule approximation for
A, denoted M , is deп¬Ѓned by
bв€’a x0 + x1 x1 + x2 xnв€’1 + xn
+ В·В·В· + f
M= f +f .
n 2 2 2
188 CHAPTER 5. THE DEFINITE INTEGRAL

Deп¬Ѓnition 5.1.4 (Trapezoidal Rule) The trapezoidal rule approximation
for A, denoted T , is deп¬Ѓned by
bв€’a 1 1 1
(f (x0 ) + f (x1 )) + (f (x1 ) + f (x2 )) + В· В· В· + (f (xnв€’1 ) + f (xn ))
T=
n 2 2 2
bв€’a 1 1
f (x0 ) + f (x1 ) + f (x2 ) + В· В· В· + f (xnв€’1 ) + f (xn ) .
=
n 2 2

Deп¬Ѓnition 5.1.5 (SimpsonвЂ™s Rule) The SimpsonвЂ™s rule approximation for
A, denoted S, is deп¬Ѓned by
bв€’a 1 x0 + x1
f (x0 ) + 4 f + f (x1 )
S=
n 6 2
1 x1 + x2
+ f (x1 ) + 4 f + f (x2 )
6 2
1 xnв€’1 + xn
+ В·В·В· + f (xnв€’1 + 4 f + f (xn )
6 2
bв€’a 1 x0 + x1 x1 + x2
В· В· f (x0 ) + 4 f
= + 2 f (x1 ) + 4 f
n 6 2 2
xnв€’1 + xn
+ В· В· В· 2 f (xnв€’1 ) + 4 f + f (xn ) .
2
Examples

Exercises 5.1
1. The sum of n terms a1 , a2 , В· В· В· , an is written in compact form in the so
called sigma notation
n
ak = a1 + a2 + В· В· В· + an .
k=1

The variable k is called the index, the number 1 is called the lower limit
n
and the number n is called the upper limit. The symbol ak is read
k=1
вЂњthe sum of ak from k = 1 to k = n.вЂќ
Verify the following sums for n = 5:
5.1. AREA APPROXIMATION 189

n
n(n + 1)
(a) k=
2
k=1
n
n(n + 1)(2n + 1)
k2 =
(b)
6
k=1
n 2
n(n + 1)
3
(c) k=
2
k=1
n
2r = 2n+1 = 1
(d)
k=1

2. Prove the following statements by using mathematical induction:
n
n(n + 1)
(a) k=
2
k=1
n
n(n + 1)(2n + 1)
k2 =
(b)
6
k=1
n 2
n(n + 1)
k3 =
(c)
2
k=1
n
2r = 2n+1 в€’ 1
(d)
k=1

3. Prove the following statements:
n n
(a) (c ak ) = c ak
k=1 k=1
n n n
(b) (ak + bk ) = ak + bk
k=1 k=1 k=1
n n n
(ak в€’ bk ) = ak в€’
(c) bk
k=1 k=1 k=1
n n n
(d) (a ak + b bk ) = a ak + b bk
k=1 k=1 k=1
190 CHAPTER 5. THE DEFINITE INTEGRAL

4. Evaluate the following sums:

6
(a) (2i)
i=0
5
1
(b)
j
j=1

4
(1 + (в€’1)k )2
(c)
k=0
5
(3m в€’ 2)
(d)
m=2

5. Let P = {a = x0 < x1 < x2 < В· В· В· < xn = b} be a partition of [a, b]
bв€’a
k, k = 0, 1, 2, В· В· В· , n. Let f (x) = x2 . Let A
such that xk = a +
n
denote the area bounded by y = f (x), y = 0, x = 0 and x = 2. Show
that
nв€’1
2
x2 .
(a) Left-hand Rule approximation of A is kв€’1
n k=1
nв€’1
2
x2 .
(b) Right-hand Rule approximation of A is k
n k=1
n 2
2 xkв€’1 + xk
(c) Mid-point Rule approximation of A is .
n 2
k=1
nв€’1
2
x2 .
(d) Trapezoidal Rule approximation of A is 2+ k
n k=1

(e) SimpsonвЂ™s Rule approximation of A

n nв€’1
2
1 xkв€’1 + xk
x2
4+4 +2 .
k
3n 2
k=1 k=1
5.1. AREA APPROXIMATION 191

In problems 6вЂ“20, use the function f , numbers a, b and n, and compute the
approximations LH, RH, M P, T, S for the area bounded by y = f (x), y =
0, x = a, x = b using the partition

bв€’a
P = {a = x0 < x1 < В· В· В· < xn = b}, where xk = a + k , and
n

n
bв€’a
(a) LH = f (xkв€’1 )
n k=1
n
bв€’a
(b) RH = f (xk )
n k=1
n
bв€’a xnв€’1 + xk
(c) M P = f
n 2
k=1
nв€’1
bв€’a 1
(d) T = f (xk ) + (f (x0 ) + f (xn ))
n 2
k=1
nв€’1 n
bв€’a xkв€’1 + xk
(e) S = (f (x0 ) + f (xn )) + 2 f (xn ) + 4 f
6n 2
k=1 k=1
1
= {LH + 4M P + RH}
6
6. f (x) = 2x, a = 0, b = 2, n = 6
1
7. f (x) = , a = 1, b = 3, n = 6
x
8. f (x) = x2 , a = 0, b = 3, n = 6

9. f (x) = x3 , a = 0, b = 2, n = 4
1
10. f (x) = , a = 0, b = 3, n = 6
1+x
1
11. f (x) = , a = 0, b = 1, n = 4
1 + x2
1
12. f (x) = в€љ , a = 0, b = 1, n = 4
2
4в€’x
192 CHAPTER 5. THE DEFINITE INTEGRAL

1
13. f (x) = , a = 0, b = 1, n = 4
4 в€’ x2
1
14. f (x) = , a = 0, b = 2, n = 4
4 + x2
1
f (x) = в€љ
15. , a = 0, b = 2, n = 4
4 + x2
в€љ
f (x) = 4 + x2 , a = 0, b = 2, n = 4
16.
в€љ
f (x) = 4 в€’ x2 , a = 0, b = 2, n = 4
17.
18. f (x) = sin x, a = 0, b = ПЂ, n = 4
ПЂ ПЂ
19. f (x) = cos x, a = в€’ , b = , n = 4
2 2
20. f (x) = sin2 x, a = 0, b = ПЂ, n = 4

5.2 The Deп¬Ѓnite Integral
Let f be a function that is continuous on a bounded and closed interval [a, b].
Let p = {a = x0 < x1 < x2 < . . . < xn = b} be a partition of [a, b], not
necessarily equally spaced. Let
mi = min{f (x) : xiв€’1 в‰¤ x в‰¤ xi }, i = 1, 2, . . . , n;
Mi = max{f (x) : xiв€’1 в‰¤ x в‰¤ xi }, i = 1, 2, . . . , n;
в€†xi = xi в€’ xiв€’1 , i = 1, 2, . . . , n;
в€† = max{в€†xi : i = 1, 2, . . . , n};
L(p) = m1 в€†x1 + m2 в€†x2 + . . . + mn в€†xn
U (p) = M1 в€†xi + M2 в€†x2 + . . . + Mn в€†xn .
We call L(p) the lower Riemann sum. We call U (p) the upper Riemann
sum. Clearly L(p) в‰¤ U (p), for every partition. Let
Lf = lub{L(p) : p is a partition of [a, b]}
Uf = glb{U (p) : p is a partition of [a, b]}.
5.2. THE DEFINITE INTEGRAL 193

Deп¬Ѓnition 5.2.1 If f is continuous on [a, b] and Lf = Uf = I, then we say
that:

(i) f is integrable on [a, b];

(ii) the deп¬Ѓnite integral of f (x) from x = a to x = b is I;

(iii) I is expressed, in symbols, by the equation
b
I= f (x)dx;
a

(iv) the symbolвЂњ вЂќ is called the вЂњintegral signвЂќ; the number вЂњaвЂќ is called
the вЂњlower limitвЂќ; the number вЂњbвЂќ is called the вЂњupper limitвЂќ; the func-
tion вЂњf (x)вЂќ is called the вЂњintegrandвЂќ; and the variable вЂњxвЂќ is called the
(dummy) вЂњvariable of integration.вЂќ

(v) If f (x) в‰Ґ 0 for each x in [a, b], then the area, A, bounded by the curves
y = f (x), y = 0, x = a and x = b, is deп¬Ѓned to be the deп¬Ѓnite integral
of f (x) from x = a to x = b. That is,
b
A= f (x)dx.
a

(vi) For convenience, we deп¬Ѓne
a a b
f (x)dx = в€’
f (x)dx = 0, f (x)dx.
a b a

Theorem 5.2.1 If a function f is continuous on a closed and bounded in-
terval [a, b], then f is integrable on [a, b].
Proof. See the proof of Theorem 5.6.3.

Theorem 5.2.2 (Linearity) Suppose that f and g are continuous on [a, b]
and c1 and c2 are two arbitrary constants. Then
194 CHAPTER 5. THE DEFINITE INTEGRAL

b b b
(i) (f (x) + g(x))dx = f (x)dx + g(x)dx
a a a
b b b
(f (x) в€’ g(x))dx = f (x)dx в€’
(ii) g(x)dx
a a a
b
b b
b
g(x)dx and
f (x)dx, c2 g(x)dx = c2
(iii) c1 f (x)dx = c1
a
a a
a
b
b
b
g(x)dx
f (x)dx + c2
(c1 f (x) + c2 g(x))dx = c1
a
a
a

Proof.
Part (i) Since f and g are continuous, f + g is continuous and hence by
Theorem 5.2.1 each of the following integrals exist:
b b b
f (x)dx, g(x)dx, and (f (x) + g(x))dx.
a a a

Let P = {a = x0 < x1 < x2 < В· В· В· < xnв€’1 < xn = b}. For each i, there exist
number c1 , c2 , c3 , d1 , d2 , and d3 on [xiв€’1 , xi ] such that
f (c1 ) = absolute minimum of f on [xiв€’1 , xi ],
g(c2 ) = absolute minimum of f on [xiв€’1 , xi ],
f (c3 ) + g(c3 ) = absolute minimum of f + g on [xiв€’1 , xi ],
f (d1 ) = absolute maximum of f on [xiв€’1 , xi ],
g(d2 ) = absolute maximum of g on [xiв€’1 , xi ],
f (d3 ) + g(d3 ) = absolute maximum of f + g on [xiв€’1 , xi ].
It follows that
f (c1 ) + g(c2 ) в‰¤ f (c3 ) + g(c3 ) в‰¤ f (d3 ) + g(d3 ) в‰¤ f (d1 ) + g(d2 )
Consequently,
Lf + Lg в‰¤ L(f +g) в‰¤ U(f +g) в‰¤ Uf + Ug (Why?)
Since f and g are integrable,
b b
Lf = Uf = f (x)dx; Lg = Ug = g(x)dx.
a a

By the squeeze principle,
b
L(f +g) = U(f +g) = (f (x) + g(x))dx
a
5.2. THE DEFINITE INTEGRAL 195

and
b b b
[f (x) + g(x)]dx = f (x)dx + g(x)dx.
a a a

This completes the proof of Part (i) of this theorem.
Part (iii) Let k be a positive constant and let F be a function that is con-
tinuous on [a, b]. Let P = {a = x0 < x1 < x2 < В· В· В· < xnв€’1 < xn = b} be
any partition of [a, b]. Then for each i there exist numbers ci and di such
that F (ci ) is the absolute minimum of F on [xiв€’1 , xi ] and F (di ) is absolute
maximum of F on [xiв€’1 , xi ]. Since k is a positive constant,
kF (ci ) = absolute minimum of kF on [xiв€’1 , xi ],
kF (di ) = absolute maximum of kF on [xiв€’1 , xi ],
в€’kF (di ) = absolute minimum of (в€’k)F on [xiв€’1 , xi ],
в€’kF (ci ) = absolute maximum of (в€’k)F on [xiв€’1 , xi ].

Then

L(P ) = F (c1 )в€†x1 + F (c2 )в€†x2 + В· В· В· + F (cn )в€†xn ,
U (P ) = F (d1 )в€†x1 + F (d2 )в€†x2 + В· В· В· + F (dn )в€†xn ,
kL(P ) = (kF )(c1 )в€†x1 + (kF )(c2 )в€†x2 + В· В· В· + (kF )(cn )в€†xn ,
kU (P ) = (kF )(d1 )в€†x1 + (kF )(d2 )в€†x2 + В· В· В· + (kF )(dn )в€†xn ,
в€’kU (P ) = (в€’kF )(d1 )в€†x1 + (в€’kF )(d2 )в€†x2 + В· В· В· + (в€’kF )(dn )в€†xn ,
в€’kL(P ) = (в€’kF )(c1 )в€†x1 + (в€’kF )(c2 )в€†x2 + В· В· В· + (в€’kF )(cn )в€†xn .

Since F is continuous, kF and (в€’k)F are both continuous and

b
Lf = Up = F (x)dx,
a
b
L(kF ) = U(kF ) = k(LF ) = k(UF ) = k F (x)dx
a
L(в€’kF ) = (в€’k)UF , U(в€’kF ) = в€’kLF ,

and hence
b
L(в€’kF ) = U(в€’kF ) = (в€’k) F (x)dx.
a
196 CHAPTER 5. THE DEFINITE INTEGRAL

Therefore,
b b b
(c1 f (x) + c2 g(x)) = c1 f (x)dx + c2 g(x)dx (Part (i))
a a a
b
b
g(x)dx (Why?)
f (x)dx + c2
= c1
a
a

This completes the proof of Part (iii) of this theorem.
Part (ii) is a special case of Part (iii) where c1 = 1 and c2 = в€’1. This
completes the proof of the theorem.

Theorem 5.2.3 (Additivity) If f is continuous on [a, b] and a < c < b,
then
b c b
f (x)dx = f (x)dx + f (x)dx.
a a c

Proof. Suppose that f is continuous on [a, b] and a < c < b. Then f is
continuous on [a, c] and on [c, b] and, hence, f is integrable on [a, b], [a, c]
and [c, b]. Let P = {a = x0 < x1 < x2 < В· В· В· xn = b}. Suppose that
xiв€’1 в‰¤ c в‰¤ xi for some i. Let P1 = {a = x0 < x1 < x2 < В· В· В· < xiв€’1 в‰¤ c}
and P2 = {c в‰¤ xi < xi+1 < В· В· В· < xn = b}. Then there exist numbers
c1 , c2 , c3 , d1 , d2 , and d3 such that
f (c1 ) = absolute minimum of f on [xiв€’1 , c],
f (d1 ) = absolute maximum of f on [xiв€’1 , c],
f (c2 ) = absolute minimum of f on [c, xi ],
f (d2 ) = absolute maximum of f on [c, xi ],
f (c3 ) = absolute minimum of f on [xiв€’1 , xi ],
f (d3 ) = absolute maximum of f on [xiв€’1 , xi ],
Also,

f (c3 ) в‰¤ f (c1 ), f (c3 ) в‰¤ f (c2 ), f (d1 ) в‰¤ f (d3 ) and f (d2 ) в‰¤ f (d3 ).

It follows that

L(P ) в‰¤ L(P1 ) + L(P2 ) в‰¤ U (P1 ) + U (P2 ) в‰¤ U (P ).

It follows that
b c b
f (x) = f (x)dx + f (x)dx.
a a c
This completes the proof of the theorem.
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