6.1-2. Equations of the Form

b
a
G(···) dt = F (x)
12.

1
0
y(t)y(xt) dt = A,0≤ x ≤ 1.
This is a special case of equation 6.2.2 with f(t)=1,a = 0, and b =1.
1
◦
. Solutions:
y
1
(x)=
√
A,
y
3
(x)=
√
A (3x – 2),
y
5
(x)=
√
A (10x
2
– 12x + 3),

y
2
(x)=–
√
A,
y
4
(x)=–
√
A (3x – 2),
y
6
(x)=–
√
A (10x
2
– 12x + 3).
2
◦
. The integral equation has some other solutions; for example,
y
7
(x)=
√
A
C

(2C +1)x
C
– C – 1


,
y
9
(x)=
√
A (ln x + 1),
y
8
(x)=–
√
A
C

(2C +1)x
C
– C – 1

,
y
10
(x)=–
√
A (ln x + 1),
where C is an arbitrary constant.
3
◦
. See 6.2.2 for some other solutions.
13.


1
0
y(t)y(xt
β
) dt = A, β >0.
1
◦
. Solutions:
y
1
(x)=
√
A,
y
3
(x)=
√
B

(β +2)x – β – 1

,
y
2
(x)=–
√
A,
y
4
(x)=–

√
B

(β +2)x – β – 1

,
where B =

2A
β(β +1)
.
2
◦
. The integral equation has some other (more complicated solutions) of the polynomial
form y(x)=
n

k=0
B
k
x
k
, where the constants B
k
can be found from the corresponding system
of algebraic equations.
14.

∞
1

y(t)y(xt) dt = Ax
–λ
, λ >0, 1≤ x < ∞.
This is a special case of equation 6.2.3 with f(t)=1,a = 1, and b = ∞.
1
◦
. Solutions:
y
1
(x)=Bx
–λ
, y
2
(x)=–Bx
–λ
, λ >
1
2
;
y
3
(x)=B

(2λ – 3)x – 2λ +2

x
–λ
, y
4
(x)=–B


(2λ – 3)x – 2λ +2

x
–λ
, λ >
3
2
;
where B =
√
A(2λ – 1).
2
◦
. For sufficiently large λ, the integral equation has some other (more complicated) solutions
of the polynomial form y(x)=
n

k=0
B
k
x
k
, where the constants B
k
can be found from the
corresponding system of algebraic equations. See 6.2.2 for some other solutions.
Page 373
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© 1998 by CRC Press LLC

15.

∞
0
e
–λt
y(t)y(xt) dt = A, λ >0, 0≤ x < ∞.
This is a special case of equation 6.2.2 with f(t)=e
–λt
, a = 0, and b = ∞.
1
◦
. Solutions:
y
1
(x)=
√
Aλ,
y
3
(x)=

1
2
Aλ (λx – 2),
y
2
(x)=–
√
Aλ,

y
4
(x)=–

1
2
Aλ (λx – 2).
2
◦
. The integral equation has some other (more complicated) solutions of the polynomial
form y(x)=
n

k=0
B
k
x
k
, where the constants B
k
can be found from the corresponding system
of algebraic equations. See 6.2.2 for some other solutions.
16.

1
0
y(t)y(x + λt) dt = A,0≤ x < ∞.
This is a special case of equation 6.2.7 with f(t) ≡ 1, a = 0, and b =1.
Solutions:
y

1
(x)=
√
A,
y
3
(x)=

3A/λ (1 – 2x),
y
2
(x)=–
√
A,
y
4
(x)=–

3A/λ (1 – 2x).
17.

∞
0
y(t)y(x + λt) dt = Ae
–βx
, A, λ, β >0, 0≤ x < ∞.
This is a special case of equation 6.2.9 with f(t) ≡ 1, a = 0, and b = ∞.
Solutions:
y
1

(x)=

Aβ(λ +1)e
–βx
,
y
3
(x)=B

β(λ +1)x – 1

e
–βx
,
y
2
(x)=–

Aβ(λ +1)e
–βx
,
y
4
(x)=–B

β(λ +1)x – 1

e
–βx
,

where B =

Aβ(λ +1)/λ.
18.

1
0
y(t)y(x – t) dt = A, –∞ < x < ∞.
This is a special case of equation 6.2.10 with f(t) ≡ 1, a = 0, and b =1.
1
◦
. Solutions with A >0:
y
1
(x)=
√
A,
y
3
(x)=
√
5A(6x
2
– 6x + 1),
y
2
(x)=–
√
A,
y

4
(x)=–
√
5A(6x
2
– 6x + 1).
2
◦
. Solutions with A <0:
y
1
(x)=
√
–3A (1 – 2x), y
2
(x)=–
√
–3A (1 – 2x).
The integral equation has some other (more complicated) solutions of the polynomial
form y(x)=
n

k=0
B
k
x
k
, where the constants B
k
can be found from the corresponding system

of algebraic equations.
19.

∞
0
e
–λt
y

x
t

y(t) dt = Ax
b
, λ >0.
Solutions: y(x)=±
√
Aλ x
b
.
Page 374
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© 1998 by CRC Press LLC
6.1-3. Equations of the Form y(x)+

b
a
K(x, t)y
2
(t) dt = F (x)

20. y(x)+A

b
a
x
λ
y
2
(t) dt =0.
Solutions:
y
1
(x)=0, y
2
(x)=–
2λ +1
A(b
2λ+1
– a
2λ+1
)
x
λ
.
21. y(x)+A

b
a
x
λ

t
µ
y
2
(t) dt =0.
Solutions:
y
1
(x)=0, y
2
(x)=–
2λ + µ +1
A(b
2λ+µ+1
– a
2λ+µ+1
)
x
λ
.
22. y(x)+A

b
a
e
–λx
y
2
(t) dt =0.
Solutions:

y
1
(x)=0, y
2
(x)=
2λ
A(e
–2λb
– e
–2λa
)
e
–λx
.
23. y(x)+A

b
a
e
–λx–µt
y
2
(t) dt =0.
Solutions:
y
1
(x)=0, y
2
(x)=
2λ + µ

A[e
–(2λ+µ)b
– e
–(2λ+µ)a
]
e
–λx
.
24. y(x)+A

b
a
x
λ
e
–µt
y
2
(t) dt =0.
This is a special case of equation 6.2.20 with f(x)=Ax
λ
and g(t)=e
–µt
.
25. y(x)+A

b
a
e
–µx

t
λ
y
2
(t) dt =0.
This is a special case of equation 6.2.20 with f(x)=Ae
–µx
and g(t)=t
λ
.
26. y(x)+A

1
0
y
2
(t) dt = Bx
µ
, µ > –1.
This is a special case of equation 6.2.22 with g(t)=A, f(x)=Bx
µ
, a = 0, and b =1.
A solution: y(x)=Bx
µ
+ λ, where λ is determined by the quadratic equation
λ
2
+
1
A


1+
2AB
µ +1

λ +
B
2
2µ +1
=0.
27. y(x)+A

b
a
t
β
y
2
(t) dt = Bx
µ
.
This is a special case of equation 6.2.22 with g(t)=At
β
and f(x)=Bx
µ
.
Page 375
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
28. y(x)+A


b
a
e
βt
y
2
(t) dt = Be
µx
.
This is a special case of equation 6.2.22 with g(t)=Ae
βt
and f(x)=Be
µx
.
29. y(x)+A

b
a
x
β
y
2
(t) dt = Bx
µ
.
This is a special case of equation 6.2.23 with g(x)=Ax
β
and f(x)=Bx
µ

.
30. y(x)+A

b
a
e
βx
y
2
(t) dt = Be
µx
.
This is a special case of equation 6.2.23 with g(x)=Ae
βx
and f(x)=Be
µx
.
6.1-4. Equations of the Form y(x)+

b
a
K(x, t)y(x)y(t) dt = F (x)
31. y(x)+A

b
a
t
β
y(x)y(t) dt = Bx
µ

.
This is a special case of equation 6.2.25 with g(t)=At
β
and f(x)=Bx
µ
.
32. y(x)+A

b
a
e
βt
y(x)y(t) dt = Be
µx
.
This is a special case of equation 6.2.25 with g(t)=Ae
βt
and f(x)=Be
µx
.
33. y(x)+A

b
a
x
β
y(x)y(t) dt = Bx
µ
.
This is a special case of equation 6.2.26 with g(x)=Ax

β
and f(x)=Bx
µ
.
34. y(x)+A

b
a
e
βx
y(x)y(t) dt = Be
µx
.
This is a special case of equation 6.2.26 with g(x)=Ae
βx
and f(x)=Be
µx
.
6.1-5. Equations of the Form y(x)+

b
a
G(···) dt = F (x)
35. y(x)+A

1
0
y(t)y(xt) dt =0.
This is a special case of equation 6.2.30 with f(t)=A, a = 0, and b =1.
1

◦
. Solutions:
y
1
(x)=–
1
A
(2C +1)x
C
, y
2
(x)=
(I
1
– I
0
)x + I
1
– I
2
I
0
I
2
– I
2
1
x
C
,

I
m
=
A
2C + m +1
, m =0,1,2,
where C is an arbitrary nonnegative constant.
There are more complicated solutions of the form y(x)=x
C
n

k=0
B
k
x
k
, where C is an
arbitrary constant and the coefficients B
k
can be found from the corresponding system of
algebraic equations.
Page 376
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
2
◦
. A solution:
y
3
(x)=

(I
1
– I
0
)x
β
+ I
1
– I
2
I
0
I
2
– I
2
1
x
C
, I
m
=
A
2C + mβ +1
, m =0,1,2,
where C and β are arbitrary constants.
There are more complicated solutions of the form y(x)=x
C
n


k=0
D
k
x
kβ
, where C and β
are arbitrary constants and the coefficients D
k
can be found from the corresponding system
of algebraic equations.
3
◦
. A solution:
y
4
(x)=
x
C
(J
1
ln x – J
2
)
J
0
J
2
– J
2
1

, J
m
=

1
0
t
2C
(ln t)
m
dt, m =0,1,2,
where C is an arbitrary constant.
There are more complicated solutions of the form y(x)=x
C
n

k=0
E
k
(ln x)
k
, where C is
an arbitrary constant and the coefficients E
k
can be found from the corresponding system of
algebraic equations.
36. y(x)+A

∞
1

y(t)y(xt) dt =0.
This is a special case of equation 6.2.30 with f(t)=A, a = 1, and b = ∞.
37. y(x)+λ

∞
1
y(t)y(xt) dt = Ax
β
.
This is a special case of equation 6.2.31 with f(t)=λ, a = 0, and b =1.
38. y(x)+A

1
0
y(t)y(x + λt) dt =0.
This is a special case of equation 6.2.35 with f(t) ≡ A, a = 0, and b =1.
1
◦
. A solution:
y(x)=
C(λ +1)
A[1 – e
C(λ+1)
]
e
Cx
,
where C is an arbitrary constant.
2
◦

. There are more complicated solutions of the form y(x)=e
Cx
n

m=0
B
m
x
m
, where C is an
arbitrary constant and the coefficients B
m
can be found from the corresponding system of
algebraic equations.
39. y(x)+A

∞
0
y(t)y(x + λt) dt =0, λ >0, 0≤ x < ∞.
This is a special case of equation 6.2.35 with f(t) ≡ A, a = 0, and b = ∞.
A solution:
y(x)=–
C(λ +1)
A
e
–Cx
,
where C is an arbitrary positive constant.
Page 377
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© 1998 by CRC Press LLC
40. y(x)+A

∞
0
e
–λt
y

x
t

y(t) dt =0, λ >0.
A solution: y(x)=–
λ
A
x
C
, where C is an arbitrary constant.
41. y(x)+A

∞
0
e
–λt
y

x
t


y(t) dt = Bx
b
, λ >0.
Solutions:
y
1
(x)=β
1
x
b
, y
2
(x)=β
2
x
b
,
where β
1
and β
2
are the roots of the quadratic equation Aβ
2
+ λβ – Bλ =0.
6.2. Equations With Quadratic Nonlinearity That Contain
Arbitrary Functions
6.2-1. Equations of the Form

b
a

G(···) dt = F (x)
1.

b
a
g(t)y(x)y(t) dt = f(x).
Solutions:
y(x)=±λf(x), λ =


b
a
f(t)g(t) dt

–1/2
.
2.

b
a
f(t)y(t)y(xt) dt = A.
1
◦
. Solutions*
y
1
(x)=

A/I
0

,
y
3
(x)=q(I
1
x – I
2
),
y
2
(x)=–

A/I
0
,
y
4
(x)=–q(I
1
x – I
2
),
where
I
m
=

b
a
t

m
f(t) dt, q =

A
I
0
I
2
2
– I
2
1
I
2

1/2
, m =0,1,2.
The integral equation has some other (more complicated) solutions of the polynomial
form y(x)=
n

k=0
B
k
x
k
, where the constants B
k
can be found from the corresponding system
of algebraic equations.

2
◦
. Solutions:
y
5
(x)=q(I
1
x
C
– I
2
), y
6
(x)=–q(I
1
x
C
– I
2
),
q =

A
I
0
I
2
2
– I
2

1
I
2

1/2
, I
m
=

b
a
t
mC
f(t) dt, m =0,1,2,
where C is an arbitrary constant.
The equation has more complicated solutions of the form y(x)=
n

k=0
B
k
x
kC
, where C is
an arbitrary constant and the coefficients B
k
can be found from the corresponding system of
algebraic equations.
* The arguments of the equations containing y(xt) in the integrand can vary, for example, within the following intervals:
(a) 0 ≤ t ≤ 1, 0 ≤ x ≤ 1 for a = 0 and b = 1; (b) 1 ≤ t < ∞,1≤ x < ∞ for a = 1 and b = ∞; (c) 0 ≤ t < ∞,0≤ x < ∞ for

a = 0 and b = ∞; or (d) a ≤ t ≤ b,0≤ x < ∞ for arbitrary a and b such that 0 ≤ a < b ≤ ∞. Case (d) is a special case of (c)
if f(t) is nonzero only on the interval a ≤ t ≤ b.
Page 378
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
3
◦
. Solutions:
y
7
(x)=p(J
0
ln x – J
1
), y
8
(x)=–p(J
0
ln x – J
1
),
p =

A
J
2
0
J
2
– J

0
J
2
1

1/2
, J
m
=

b
a
(ln t)
m
f(t) dt.
The equation has more complicated solutions of the form y(x)=
n

k=0
E
k
(ln x)
k
, where the
constants E
k
can be found from the corresponding system of algebraic equations.
3.

b

a
f(t)y(t)y(xt) dt = Ax
β
.
1
◦
. Solutions:
y
1
(x)=

A/I
0
x
β
,
y
3
(x)=q(I
1
x – I
2
) x
β
,
y
2
(x)=–

A/I

0
x
β
,
y
4
(x)=–q(I
1
x – I
2
) x
β
,
where
I
m
=

b
a
t
2β+m
f(t) dt, q =

A
I
2
(I
0
I

2
– I
2
1
)
, m =0,1,2.
2
◦
. The substitution y(x)=x
β
w(x) leads to an equation of the form 6.2.2:

b
a
g(t)w(t)w(xt) dt = A, g(x)=f(x)x
2β
.
Therefore, the integral equation in question has more complicated solutions.
4.

b
a
f(t)y(t)y(xt) dt = A ln x + B.
This equation has solutions of the form y(x)=p ln x +q. The constants p and q are determined
from the following system of two second-order algebraic equations:
I
1
p
2
+ I

0
pq = A, I
2
p
2
+2I
1
pq + I
0
q
2
= B,
where
I
m
=

b
a
f(t)(ln t)
m
dt, m =0,1,2.
5.

b
a
f(t)y(t)y(xt) dt = Ax
λ
ln x + Bx
λ

.
The substitution y(x)=x
λ
w(x) leads to an equation of the form 6.2.4:

b
a
g(t)w(t)w(xt) dt = A ln x + B, g(t)=f (t)t
2λ
.
6.

∞
0
f(t)y(t)y

x
t

dt = Ax
λ
.
Solutions:
y
1
(x)=

A
I
x

λ
, y
2
(x)=–

A
I
x
λ
, I =

∞
0
f(t) dt.
Page 379
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© 1998 by CRC Press LLC
7.

b
a
f(t)y(t)y(x + λt) dt = A, λ >0.
1
◦
. Solutions*
y
1
(x)=

A/I

0
,
y
3
(x)=q(I
0
x – I
1
),
y
2
(x)=–

A/I
0
,
y
4
(x)=–q(I
0
x – I
1
),
where
I
m
=

b
a

t
m
f(t) dt, q =

A
λ(I
2
0
I
2
– I
0
I
2
1
)
, m =0,1,2.
2
◦
. The integral equation has some other (more complicated) solutions of the polynomial
form y(x)=
n

k=0
B
k
x
k
, where the constants B
k

can be found from the corresponding system
of algebraic equations.
8.

b
a
f(t)y(t)y(x + λt) dt = Ax + B, λ >0.
A solution: y(x)=βx + µ, where the constants β and µ are determined from the following
system of two second-order algebraic equations:
I
0
βµ + I
1
β
2
= A, I
0
µ
2
+(λ +1)I
1
βµ + λI
2
β
2
= B, I
m
=

b

a
t
m
f(t) dt. (1)
Multiplying the first equation by B and the second by –A and adding the resulting equations,
we obtain the quadratic equation
AI
0
z
2
+

(λ +1)AI
1
– BI
0

z + λAI
2
– BI
1
=0, z = µ/β. (2)
In general, to each root of equation (2) two solutions of system (1) correspond. Therefore,
the original integral equation can have at most four solutions of this form. If the discriminant
of equation (2) is negative, then the integral equation has no such solutions.
The integral equation has some other (more complicated) solutions of the polynomial
form y(x)=
n

k=0

β
k
x
k
, where the constants β
k
can be found from the corresponding system
of algebraic equations.
9.

b
a
f(t)y(t)y(x + λt) dt = Ae
–βx
, λ >0.
1
◦
. Solutions:
y
1
(x)=

A/I
0
e
–βx
,
y
3
(x)=q(I

0
x – I
1
)e
–βx
,
y
2
(x)=–

A/I
0
e
–βx
,
y
4
(x)=–q(I
0
x – I
1
)e
–βx
,
where
I
m
=

b

a
t
m
e
–β(λ+1)t
f(t) dt, q =

A
λ(I
2
0
I
2
– I
0
I
2
1
)
, m =0,1,2.
2
◦
. The equation has more complicated solutions of the form y(x)=e
–βx
n

k=0
B
k
x

k
, where
the constants B
k
can be found from the corresponding system of algebraic equations.
* The arguments of the equations containing y(x+λt) in the integrand can vary within the following intervals: (a) 0 ≤t < ∞,
0 ≤ x < ∞ for a = 0 and b = ∞ or (b) a ≤ t ≤ b,0≤ x < ∞ for arbitrary a and b such that 0 ≤ a < b < ∞. Case (b) is a
special case of (a) if f(t) is nonzero only on the interval a ≤ t ≤ b.
Page 380
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
3
◦
. The substitution y(x)=e
–βx
w(x) leads to an equation of the form 6.2.7:

b
a
e
–β(λ+1)t
f(t)w(t)w(x + λt) dt = A.
10.

b
a
f(t)y(t)y(x – t) dt = A.
1
◦
. Solutions*

y
1
(x)=

A/I
0
,
y
3
(x)=q(I
0
x – I
1
),
y
2
(x)=–

A/I
0
,
y
4
(x)=–q(I
0
x – I
1
),
where
I

m
=

b
a
t
m
f(t) dt, q =

A
I
0
I
2
1
– I
2
0
I
2
, m =0,1,2.
2
◦
. The integral equation has some other (more complicated) solutions of the polynomial
form y(x)=
n

k=0
λ
k

x
k
, where the constants λ
k
can be found from the corresponding system
of algebraic equations. For n = 3, such a solution is presented in 6.1.18.
11.

b
a
f(t)y(t)y(x – t) dt = Ax + B.
A solution: y(x)=λx + µ, where the constants λ and µ are determined from the following
system of two second-order algebraic equations:
I
0
λµ + I
1
λ
2
= A, I
0
µ
2
– I
2
λ
2
= B, I
m
=


b
a
t
m
f(t) dt, m = 0, 1, 2. (1)
Multiplying the first equation by B and the second by –A and adding the results, we obtain
the quadratic equation
AI
0
z
2
– BI
0
z – AI
2
– BI
1
=0, z = µ/λ. (2)
In general, to each root of equation (2) two solutions of system (1) correspond. Therefore,
the original integral equation can have at most four solutions of this form. If the discriminant
of equation (2) is negative, then the integral equation has no such solutions.
The integral equation has some other (more complicated) solutions of the polynomial
form y(x)=
n

k=0
λ
k
x

k
, where the constants λ
k
can be found from the corresponding system
of algebraic equations.
12.

b
a
f(t)y(t)y(x – t) dt =
n

k=0
A
k
x
k
.
This equation has solutions of the form
y(x)=
n

k=0
λ
k
x
k
, (1)
where the constants λ
k

are determined from the system of algebraic equations obtained by
substituting solution (1) into the original integral equation and matching the coefficients of
like powers of x.
* The arguments of the equations containing y(x–t) in the integrand can vary within the following intervals: (a) –∞ < t< ∞,
–∞ < x < ∞ for a = –∞ and b = ∞ or (b) a ≤ t ≤ b, –∞ ≤ x < ∞, for arbitrary a and b such that –∞ < a < b < ∞.
Case (b) is a special case of (a) if f(t) is nonzero only on the interval a ≤ t ≤ b.
Page 381
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© 1998 by CRC Press LLC
13.

b
a
f(t)y(x – t)y(t) dt = Ae
λx
.
Solutions:
y
1
(x)=

A/I
0
e
λx
,
y
3
(x)=q(I
0

x – I
1
)e
λx
,
y
2
(x)=–

A/I
0
e
λx
,
y
4
(x)=–q(I
0
x – I
1
)e
λx
,
where
I
m
=

b
a

t
m
f(t) dt, q =

A
I
0
I
2
1
– I
2
0
I
2
, m =0,1,2.
The integral equation has more complicated solutions of the form y(x)=e
λx
n

k=0
B
k
x
k
, where
the constants B
k
can be found from the corresponding system of algebraic equations.
14.


b
a
f(t)y(t)y(x – t) dt = A sinh λx.
A solution:
y(x)=p sinh λx + q cosh λx. (1)
Here p and q are roots of the algebraic system
I
0
pq + I
cs
(p
2
– q
2
)=A, I
cc
q
2
– I
ss
p
2
= 0, (2)
where the notation
I
0
=

b

a
f(t) dt, I
cs
=

b
a
f(t) cosh(λt) sinh(λt) dt,
I
cc
=

b
a
f(t) cosh
2
(λt) dt, I
ss
=

b
a
f(t) sinh
2
(λt) dt
is used. Different solutions of system (2) generate different solutions (1) of the integral
equation.
It follows from the second equation of (2) that q = ±

I

ss
/I
cc
p. Using this expression to
eliminate q from the first equation of (2), we obtain the following four solutions:
y
1,2
(x)=p

sinh λx ± k cosh λx

, y
3,4
(x)=–p

sinh λx ± k cosh λx

,
k =

I
ss
I
cc
, p =

A
(1 – k
2
)I

cs
± kI
0
.
15.

b
a
f(t)y(t)y(x – t) dt = A cosh λx.
A solution:
y(x)=p sinh λx + q cosh λx. (1)
Here p and q are roots of the algebraic system
I
0
pq + I
cs
(p
2
– q
2
)=0, I
cc
q
2
– I
ss
p
2
= A, (2)
where we use the notation introduced in 6.2.14. Different solutions of system (2) generate

different solutions (1) of the integral equation.
Page 382
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© 1998 by CRC Press LLC
16.

b
a
f(t)y(t)y(x – t) dt = A sin λx.
A solution:
y(x)=p sin λx + q cos λx. (1)
Here p and q are roots of the algebraic system
I
0
pq + I
cs
(p
2
+ q
2
)=A, I
cc
q
2
– I
ss
p
2
= 0, (2)
where

I
0
=

b
a
f(t) dt, I
cs
=

b
a
f(t) cos(λt) sin(λt) dt,
I
cc
=

b
a
f(t) cos
2
(λt) dt, I
ss
=

b
a
f(t) sin
2
(λt) dt.

It follows from the second equation of (2) that q = ±

I
ss
/I
cc
p. Using this expression to
eliminate q from the first equation of (2), we obtain the following four solutions:
y
1,2
(x)=p

sin λx ± k cos λx

, y
3,4
(x)=–p

sin λx ± k cos λx

,
k =

I
ss
I
cc
, p =

A

(1 + k
2
)I
cs
± kI
0
.
17.

b
a
f(t)y(t)y(x – t) dt = A cos λx.
A solution:
y(x)=p sin λx + q cos λx. (1)
Here p and q are roots of the algebraic system
I
0
pq + I
cs
(p
2
+ q
2
)=0, I
cc
q
2
– I
ss
p

2
= A, (2)
where we use the notation introduced in 6.2.16. Different solutions of system (2) generate
different solutions (1) of the integral equation.
18.

1
0
y(t)y(ξ) dt = A, ξ = f(x)t.
1
◦
. Solutions:
y
1
(t)=
√
A,
y
3
(t)=
√
A (3t – 2),
y
5
(t)=
√
A (10t
2
– 12t + 3),
y

2
(t)=–
√
A,
y
4
(t)=–
√
A (3t – 2),
y
6
(t)=–
√
A (10t
2
– 12t + 3).
2
◦
. The integral equation has some other (more complicated) solutions of the polynomial
form y(t)=
n

k=0
B
k
t
k
, where the constants B
k
can be found from the corresponding system of

algebraic equations.
3
◦
. The substitution z = f(x) leads to an equation of the form 6.1.12.
Page 383
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© 1998 by CRC Press LLC
6.2-2. Equations of the Form y(x)+

b
a
K(x, t)y
2
(t) dt = F (x)
19. y(x)+

b
a
f(x)y
2
(t) dt =0.
Solutions: y
1
(x)=0andy
2
(x)=λf(x), where λ = –


b
a

f
2
(t) dt

–1
.
20. y(x)+

b
a
f(x)g(t)y
2
(t) dt =0.
This is a special case of equation 6.8.29.
Solutions: y
1
(x)=0andy
2
(x)=λf(x), where λ = –


b
a
f
2
(t)g(t) dt

–1
.
21. y(x)+A


b
a
y
2
(t) dt = f(x).
This is a special case of equation 6.8.27.
A solution: y(x)=f(x)+λ, where λ is determined by the quadratic equation
A(b – a)λ
2
+(1+2AI
1
)λ + AI
2
= 0, where I
1
=

b
a
f(t) dt, I
2
=

b
a
f
2
(t) dt.
22. y(x)+


b
a
g(t)y
2
(t) dt = f(x).
This is a special case of equation 6.8.29.
A solution: y(x)=f(x)+λ, where λ is determined by the quadratic equation
I
0
λ
2
+(1+2I
1
)λ + I
2
= 0, where I
m
=

b
a
f
m
(t)g(t) dt, m =0,1,2.
23. y(x)+

b
a
g(x)y

2
(t) dt = f(x).
Solution: y(x)=λg(x)+f(x), where λ is determined by the quadratic equation
I
gg
λ
2
+(1+2I
fg
)λ + I
ff
=0,
I
gg
=

b
a
g
2
(t) dt, I
fg
=

b
a
f(t)g(t) dt, I
ff
=


b
a
f
2
(t) dt.
24. y(x)+

b
a

g
1
(x)h
1
(t)+g
2
(x)h
2
(t)

y
2
(t) dt = f(x).
A solution: y(x)=λ
1
g
1
(x)+λ
2
g

2
(x)+f(x), where the constants λ
1
and λ
2
can be found
from a system of two second-order algebraic equations (this system can be obtained from the
more general system presented in 6.8.42).
Page 384
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
6.2-3. Equations of the Form y(x)+

b
a

K
nm
(x, t)y
n
(x)y
m
(t) dt = F (x), n + m ≤ 2
25. y(x)+

b
a
g(t)y(x)y(t) dt = f(x).
Solutions:
y

1
(x)=λ
1
f(x), y
2
(x)=λ
2
f(x),
where λ
1
and λ
2
are the roots of the quadratic equation
Iλ
2
+ λ – 1=0, I =

b
a
f(t)g(t) dt.
26. y(x)+

b
a
g(x)y(x)y(t) dt = f(x).
A solution:
y(x)=
f(x)
1+λg(x)
,

where λ is a root of the algebraic (or transcendental) equation
λ –

b
a
f(t) dt
1+λg(t)
=0.
Different roots generate different solutions of the integral equation.
27. y(x)+

b
a

g
1
(t)y
2
(x)+g
2
(x)y(t)

dt = f(x).
Solution in an implicit form:
y(x)+Iy
2
(x)+λg
2
(x) – f(x)=0, I =


b
a
g
1
(t) dt, (1)
where λ is determined by the algebraic equation
λ =

b
a
y(t) dt. (2)
Here the function y(x)=y(x, λ) obtained by solving the quadratic equation (1) must be
substituted in the integrand of (2).
28. y(x)+

b
a

g
1
(t)y
2
(x)+g
2
(x)y
2
(t)

dt = f(x).
Solution in an implicit form:

y(x)+Iy
2
(x)+λg
2
(x) – f(x)=0, I =

b
a
g
1
(t) dt, (1)
where λ is determined by the algebraic equation
λ =

b
a
y
2
(t) dt. (2)
Here the function y(x)=y(x, λ) obtained by solving the quadratic equation (1) must be
substituted into the integrand of (2).
29. y(x)+

b
a

g
11
(x)h
11

(t)y
2
(x)+g
12
(x)h
12
(t)y(x)y(t)+g
22
(x)h
22
(t)y
2
(t)
+ g
1
(x)h
1
(t)y(x)+g
2
(x)h
2
(t)y(t)

dt = f(x).
This is a special case of equation 6.8.44.
Page 385
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
6.2-4. Equations of the Form y(x)+


b
a
G(···) dt = F (x)
30. y(x)+

b
a
f(t)y(t)y(xt) dt =0.
1
◦
. Solutions:
y
1
(x)=–
1
I
0
x
C
, y
2
(x)=
(I
1
– I
0
)x + I
1
– I
2

I
0
I
2
– I
2
1
x
C
,
I
m
=

b
a
f(t)t
2C+m
dt, m =0,1,2,
where C is an arbitrary constant.
There are more complicated solutions of the form y(x)=x
C
n

k=0
B
k
x
k
, where C is an

arbitrary constant and the coefficients B
k
can be found from the corresponding system of
algebraic equations.
2
◦
. A solution:
y
3
(x)=
(I
1
– I
0
)x
β
+ I
1
– I
2
I
0
I
2
– I
2
1
x
C
,

I
m
=

b
a
f(t)t
2C+mβ
dt, m =0,1,2,
where C and β are arbitrary constants.
There are more complicated solutions of the form y(x)=x
C
n

k=0
D
k
x
kβ
, where C and β
are arbitrary constants and the coefficients D
k
can be found from the corresponding system
of algebraic equations.
3
◦
. A solution:
y
4
(x)=

x
C
(J
1
ln x – J
2
)
J
0
J
2
– J
2
1
,
J
m
=

b
a
f(t)t
2C
(ln t)
m
dt, m =0,1,2,
where C is an arbitrary constant.
There are more complicated solutions of the form y(x)=x
C
n


k=0
E
k
(ln x)
k
, where C is
an arbitrary constant and the coefficients E
k
can be found from the corresponding system of
algebraic equations.
4
◦
. The equation also has the trivial solution y(x) ≡ 0.
5
◦
. The substitution y(x)=x
β
w(x) leads to an equation of the same form,
w(x)+

b
a
g(t)w(t)w(xt) dt =0, g(x)=f(x)x
2β
.
Page 386
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© 1998 by CRC Press LLC
31. y(x)+


b
a
f(t)y(t)y(xt) dt = Ax
β
.
1
◦
. Solutions:
y
1
(x)=k
1
x
β
, y
2
(x)=k
2
x
β
,
where k
1
and k
2
are the roots of the quadratic equation
Ik
2
+ k – A =0, I =


b
a
f(t)t
2β
dt.
2
◦
. Solutions:
y(x)=x
β
(λx + µ),
where λ and µ are determined from the following system of two algebraic equations (this
system can be reduced to a quadratic equation):
I
2
λ + I
1
µ +1=0, I
1
λµ + I
0
µ
2
+ µ – A =0
where I
m
=

b

a
f(t)t
2β+m
dt, m =0,1,2.
3
◦
. There are more complicated solutions of the form y(x)=x
β
n

m=0
B
m
x
m
, where the B
m
can be found from the corresponding system of algebraic equations.
32. y(x)+

b
a
f(t)y(t)y(xt) dt = A ln x + B.
This equation has solutions of the form y(x)=p ln x + q, where the constants p and q can be
found from a system of two second-order algebraic equations.
33. y(x)+

∞
0
f(t)y(t)y


x
t

dt =0.
1
◦
. A solution:
y(x)=–kx
C
, k =


∞
0
f(t) dt

–1
,
where C is an arbitrary constant.
2
◦
. The equation has the trivial solution y(x) ≡ 0.
3
◦
. The substitution y(x)=x
β
w(x) leads to an equation of the same form,
w(x)+


∞
0
f(t)w(t)w

x
t

dt =0.
34. y(x)+

∞
0
f(t)y

x
t

y(t) dt = Ax
b
.
Solutions:
y
1
(x)=λ
1
x
b
, y
2
(x)=λ

2
x
b
,
where λ
1
and λ
2
are the roots of the quadratic equation
Iλ
2
+ λ – A =0, I =

∞
0
f(t) dt.
Page 387
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
35. y(x)+

b
a
f(t)y(t)y(x + λt) dt =0, λ >0.
1
◦
. Solutions:
y
1
(x)=–

1
I
0
exp(–Cx), y
2
(x)=
I
2
– I
1
x
I
2
1
– I
0
I
2
exp(–Cx),
I
m
=

b
a
t
m
exp

–C(λ +1)t


f(t) dt, m =0,1,2,
where C is an arbitrary constant.
2
◦
. There are more complicated solutions of the form y(x)=exp(–Cx)
n

k=0
A
k
x
k
, where C
is an arbitrary constant and the coefficients A
k
can be found from the corresponding system
of algebraic equations.
3
◦
. The equation also has the trivial solution y(x) ≡ 0.
4
◦
. The substitution y(x)=e
βx
w(x) leads to a similar equation:
w(x)+

b
a

g(t)w(t)w(x + λt) dt =0, g(t)=e
β(λ+1)t
f(t).
36. y(x)+

b
a
f(t)y(x + λt)y(t) dt = Ae
–µx
, λ >0.
1
◦
. Solutions:
y
1
(x)=k
1
e
–µx
, y
2
(x)=k
2
e
–µx
,
where k
1
and k
2

are the roots of the quadratic equation
Ik
2
+ k – A =0, I =

b
a
e
–µ(λ+1)t
f(t) dt.
2
◦
. There are more complicated solutions of the form y(x)=e
–µx
n

m=0
B
m
x
m
, where the B
m
can be found from the corresponding system of algebraic equations.
3
◦
. The substitution y(x)=e
βx
w(x) leads to an equation of the same form,
w(x)+


b
a
g(t)w(t)w(x – t) dt = Ae
(λ–β)x
, g(t)=f(t)e
β(λ+1)t
.
37. y(x)+

b
a
f(t)y(t)y(x – t) dt =0.
1
◦
. Solutions:
y
1
(x)=–
1
I
0
exp(Cx), y
2
(x)=
I
2
– I
1
x

I
2
1
– I
0
I
2
exp(Cx), I
m
=

b
a
t
m
f(t) dt,
where C is an arbitrary constant and m =0,1,2.
2
◦
. There are more complicated solutions of the form y(x)=exp(Cx)
n

k=0
A
k
x
k
, where C is
an arbitrary constant and the coefficients A
k

can be found from the corresponding system of
algebraic equations.
3
◦
. The equation also has the trivial solution y(x) ≡ 0.
4
◦
. The substitution y(x)=exp(Cx)w(x) leads to an equation of the same form:
w(x)+

b
a
f(t)w(t)w(x – t) dt =0.
Page 388
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38. y(x)+

b
a
f(t)y(x – t)y(t) dt = Ae
λx
.
1
◦
. Solutions:
y
1
(x)=k
1

e
λx
, y
2
(x)=k
2
e
λx
,
where k
1
and k
2
are the roots of the quadratic equation
Ik
2
+ k – A =0, I =

b
a
f(t) dt.
2
◦
. The substitution y(x)=e
βx
w(x) leads to an equation of the same form,
w(x)+

b
a

f(t)w(t)w(x – t) dt = Ae
(λ–β)x
.
39. y(x)+

b
a
f(t)y(t)y(x – t) dt = A sinh λx.
A solution:
y(x)=p sinh λx + q cosh λx. (1)
Here p and q are roots of the algebraic system
p + I
0
pq + I
cs
(p
2
– q
2
)=A, q + I
cc
q
2
– I
ss
p
2
= 0, (2)
where
I

0
=

b
a
f(t) dt, I
cs
=

b
a
f(t) cosh(λt) sinh(λt) dt,
I
cc
=

b
a
f(t) cosh
2
(λt) dt, I
ss
=

b
a
f(t) sinh
2
(λt) dt.
Different solutions of system (2) generate different solutions (1) of the integral equation.

40. y(x)+

b
a
f(t)y(t)y(x – t) dt = A cosh λx.
A solution:
y(x)=p sinh λx + q cosh λx. (1)
Here p and q are roots of the algebraic system
p + I
0
pq + I
cs
(p
2
– q
2
)=0, q + I
cc
q
2
– I
ss
p
2
= A, (2)
where we use the notation introduced in 6.2.39. Different solutions of system (2) generate
different solutions (1) of the integral equation.
41. y(x)+

b

a
f(t)y(t)y(x – t) dt = A sin λx.
A solution:
y(x)=p sin λx + q cos λx. (1)
Here p and q are roots of the algebraic system
p + I
0
pq + I
cs
(p
2
+ q
2
)=A, q + I
cc
q
2
– I
ss
p
2
= 0, (2)
where
I
0
=

b
a
f(t) dt, I

cs
=

b
a
f(t) cos(λt) sin(λt) dt,
I
cc
=

b
a
f(t) cos
2
(λt) dt, I
ss
=

b
a
f(t) sin
2
(λt) dt.
Different solutions of system (2) generate different solutions (1) of the integral equation.
Page 389
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42. y(x)+

b

a
f(t)y(t)y(x – t) dt = A cos λx.
A solution:
y(x)=p sin λx + q cos λx. (1)
Here p and q are roots of the algebraic system
p + I
0
pq + I
cs
(p
2
+ q
2
)=0, q + I
cc
q
2
– I
ss
p
2
= A, (2)
where we use the notation introduced in 6.2.41. Different solutions of system (2) generate
different solutions (1) of the integral equation.
6.3. Equations With Power-Law Nonlinearity
6.3-1. Equations of the Form

b
a
G(···) dt = F (x)

1.

b
a
t
λ
y
µ
(x)y
β
(t) dt = f(x).
A solution:
y(x)=A

f(x)

1
µ
, A =


b
a
t
λ

f(t)

β
µ

dt

–
1
µ+β
.
2.

b
a
e
λt
y
µ
(x)y
β
(t) dt = f(x).
A solution:
y(x)=A

f(x)

1
µ
, A =


b
a
e

λt

f(t)

β
µ
dt

–
1
µ+β
.
3.

∞
0
f(x
a
t)t
b
y

x
k
t

y(t)

s
dt = Ax

c
.
A solution:
y(x)=

A
I

1
s+1
x
λ
, λ =
a + c + ab
k – a – as
,
I =

∞
0
f(t)t
β
dt, β =
a + c + as + bk + cs
k – a – as
.
6.3-2. Equations of the Form y(x)+

b
a

K(x, t)y
β
(t) dt = F (x)
4. y(x)+A

b
a
t
λ
y
β
(t) dt = g(x).
This is a special case of equation 6.8.27 with f(t, y)=At
λ
y
β
.
5. y(x)+A

b
a
e
µt
y
β
(t) dt = g(x).
This is a special case of equation 6.8.27 with f(t, y)=Ae
µt
y
β

.
Page 390
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6. y(x)+A

b
a
e
λ(x–t)
y
β
(t) dt = g(x).
This is a special case of equation 6.8.28 with f(t, y)=Ay
β
.
7. y(x) –

b
a
g(x)y
β
(t) dt =0.
A solution:
y(x)=λg(x), λ =


b
a
g

β
(t) dt

1
1–β
.
For β > 0, the equation also has the trivial solution y(x) ≡ 0.
8. y(x) –

b
a
g(x)y
β
(t) dt = h(x).
This is a special case of equation 6.8.29 with f(t, y)=–y
β
.
9. y(x)+A

b
a
cosh(λx + µt)y
β
(t) dt = h(x).
This is a special case of equation 6.8.31 with f(t, y)=Ay
β
.
10. y(x)+A

b

a
sinh(λx + µt)y
β
(t) dt = h(x).
This is a special case of equation 6.8.32 with f(t, y)=Ay
β
.
11. y(x)+A

b
a
cos(λx + µt)y
β
(t) dt = h(x).
This is a special case of equation 6.8.33 with f(t, y)=Ay
β
.
12. y(x)+A

b
a
sin(λx + µt)y
β
(t) dt = h(x).
This is a special case of equation 6.8.34 with f(t, y)=Ay
β
.
13. y(x)+

∞

0
f

t
x


y(t) dt = Ax
2
.
Solutions: y
k
(x)=β
2
k
x
2
, where β
k
(k = 1, 2) are the roots of the quadratic equations
β
2
± Iβ – A =0, I =

∞
0
zf(z) dz.
14. y(x) –

∞

0
t
λ
f

t
x


y(t)

β
dt =0, β ≠ 1.
A solution:
y(x)=Ax
1+λ
1–β
, A
1–β
=

∞
0
z
λ+β
1–β
f(z) dz.
15. y(x) –

∞

–∞
e
λt
f(ax + bt)

y(t)

β
dt =0, b ≠ 0, aβ ≠ –b.
A solution:
y(x)=A exp

–
aλ
aβ + b
x

, A
1–β
=

∞
–∞
exp

λb
aβ + b
z

f(bz) dz.

Page 391
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6.3-3. Equations of the Form y(x)+

b
a
G(···) dt = F (x)
16. y(x)+A

b
a
y
β
(x)y
µ
(t) dt = f(x).
Solution in an implicit form:
y(x)+Aλy
β
(x) – f(x)=0, (1)
where λ is determined by the algebraic (or transcendental) equation
λ =

b
a
y
µ
(t) dt. (2)
Here the function y(x)=y(x, λ) obtained by solving the quadratic equation (1) must be

substituted in the integrand of (2).
17. y(x)+

b
a
g(t)y(x)y
µ
(t) dt = f(x).
A solution: y(x)=λf(x), where λ is determined from the algebraic (or transcendental)
equation
Iλ
µ+1
+ λ – 1=0, I =

b
a
g(t)f
µ
(t) dt.
18. y(x)+

b
a
g(x)y(x)y
µ
(t) dt = f(x).
A solution:
y(x)=
f(x)
1+λg(x)

,
where λ is a root of the algebraic (or transcendental) equation
λ –

b
a
f
µ
(t) dt
[1 + λg(t)]
µ
=0.
Different roots generate different solutions of the integral equation.
19. y(x)+

b
a

g
1
(t)y
2
(x)+g
2
(x)y
µ
(t)

dt = f(x).
Solution in an implicit form:

y(x)+Iy
2
(x)+λg
2
(x) – f(x)=0, I =

b
a
g
1
(t) dt, (1)
where λ is determined by the algebraic (or transcendental) equation
λ =

b
a
y
µ
(t) dt. (2)
Here the function y(x)=y(x, λ) obtained by solving the quadratic equation (1) must be
substituted in the integrand of (2).
20. y(x)+

b
a

g
1
(x)h
1

(t)y
k
(x)y
s
(t)+g
2
(x)h
2
(t)y
p
(x)y
q
(t)

dt = f(x).
This is a special case of equation 6.8.44.
21. y(x)+A

b
a
y(xt)y
β
(t) dt =0.
This is a special case of equation 6.8.45 with f(t, y)=Ay
β
.
Page 392
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6.4. Equations With Exponential Nonlinearity

6.4-1. Integrands With Nonlinearity of the Form exp[βy(t)]
1. y(x)+A

b
a
exp[βy(t)] dt = g(x).
This is a special case of equation 6.8.27 with f(t, y)=A exp(βy).
2. y(x)+A

b
a
t
µ
exp[βy(t)] dt = g(x).
This is a special case of equation 6.8.27 with f(t, y)=At
µ
exp(βy).
3. y(x)+A

b
a
exp

µt + βy(t)

dt = g(x).
This is a special case of equation 6.8.27 with f(t, y)=A exp(µt) exp(βy).
4. y(x)+A

b

a
exp

λ(x – t)+βy(t)

dt = g(x).
This is a special case of equation 6.8.28 with f(t, y)=A exp(βy).
5. y(x)+

b
a
g(x) exp[βy(t)] dt = h(x).
This is a special case of equation 6.8.29 with f(t, y) = exp(βy).
6. y(x)+A

b
a
cosh(λx + µt) exp[βy(t)] dt = h(x).
This is a special case of equation 6.8.31 with f(t, y)=A exp(βy).
7. y(x)+A

b
a
sinh(λx + µt) exp[βy(t)] dt = h(x).
This is a special case of equation 6.8.32 with f(t, y)=A exp(βy).
8. y(x)+A

b
a
cos(λx + µt) exp[βy(t)] dt = h(x).

This is a special case of equation 6.8.33 with f(t, y)=A exp(βy).
9. y(x)+A

b
a
sin(λx + µt) exp[βy(t)] dt = h(x).
This is a special case of equation 6.8.34 with f(t, y)=A exp(βy).
6.4-2. Other Integrands
10. y(x)+A

b
a
exp

βy(x)+γy(t)

dt = h(x).
This is a special case of equation 6.8.43 with g(x, y)=A exp(βy) and f(t, y) = exp(γy).
11. y(x)+A

b
a
y(xt) exp[βy(t)] dt =0.
This is a special case of equation 6.8.45 with f(t, y)=A exp(βy).
Page 393
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6.5. Equations With Hyperbolic Nonlinearity
6.5-1. Integrands With Nonlinearity of the Form cosh[βy(t)]
1. y(x)+A


b
a
cosh[βy(t)] dt = g(x).
This is a special case of equation 6.8.27 with f(t, y)=A cosh(βy).
2. y(x)+A

b
a
t
µ
cosh
k
[βy(t)] dt = g(x).
This is a special case of equation 6.8.27 with f(t, y)=At
µ
cosh
k
(βy).
3. y(x)+A

b
a
cosh(µt) cosh[βy(t)] dt = g(x).
This is a special case of equation 6.8.27 with f(t, y)=A cosh(µt) cosh(βy).
4. y(x)+A

b
a
e

λ(x–t)
cosh[βy(t)] dt = g(x).
This is a special case of equation 6.8.28 with f(t, y)=A cosh(βy).
5. y(x)+

b
a
g(x) cosh[βy(t)] dt = h(x).
This is a special case of equation 6.8.29 with f(t, y) = cosh(βy).
6. y(x)+A

b
a
cosh(λx + µt) cosh[βy(t)] dt = h(x).
This is a special case of equation 6.8.31 with f(t, y)=A cosh(βy).
7. y(x)+A

b
a
sinh(λx + µt) cosh[βy(t)] dt = h(x).
This is a special case of equation 6.8.32 with f(t, y)=A cosh(βy).
8. y(x)+A

b
a
cos(λx + µt) cosh[βy(t)] dt = h(x).
This is a special case of equation 6.8.33 with f(t, y)=A cosh(βy).
9. y(x)+A

b

a
sin(λx + µt) cosh[βy(t)] dt = h(x).
This is a special case of equation 6.8.34 with f(t, y)=A cosh(βy).
6.5-2. Integrands With Nonlinearity of the Form sinh[βy(t)]
10. y(x)+A

b
a
sinh[βy(t)] dt = g(x).
This is a special case of equation 6.8.27 with f(t, y)=A sinh(βy).
Page 394
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11. y(x)+A

b
a
t
µ
sinh
k
[βy(t)] dt = g(x).
This is a special case of equation 6.8.27 with f(t, y)=At
µ
sinh
k
(βy).
12. y(x)+A

b

a
sinh(µt) sinh[βy(t)] dt = g(x).
This is a special case of equation 6.8.27 with f(t, y)=A sinh(µt) sinh(βy).
13. y(x)+A

b
a
e
λ(x–t)
sinh[βy(t)] dt = g(x).
This is a special case of equation 6.8.28 with f(t, y)=A sinh(βy).
14. y(x)+

b
a
g(x) sinh[βy(t)] dt = h(x).
This is a special case of equation 6.8.29 with f(t, y) = sinh(βy).
15. y(x)+A

b
a
cosh(λx + µt) sinh[βy(t)] dt = h(x).
This is a special case of equation 6.8.31 with f(t, y)=A sinh(βy).
16. y(x)+A

b
a
sinh(λx + µt) sinh[βy(t)] dt = h(x).
This is a special case of equation 6.8.32 with f(t, y)=A sinh(βy).
17. y(x)+A


b
a
cos(λx + µt) sinh[βy(t)] dt = h(x).
This is a special case of equation 6.8.33 with f(t, y)=A sinh(βy).
18. y(x)+A

b
a
sin(λx + µt) sinh[βy(t)] dt = h(x).
This is a special case of equation 6.8.34 with f(t, y)=A sinh(βy).
6.5-3. Integrands With Nonlinearity of the Form tanh[βy(t)]
19. y(x)+A

b
a
tanh[βy(t)] dt = g(x).
This is a special case of equation 6.8.27 with f(t, y)=A tanh(βy).
20. y(x)+A

b
a
t
µ
tanh
k
[βy(t)] dt = g(x).
This is a special case of equation 6.8.27 with f(t, y)=At
µ
tanh

k
(βy).
21. y(x)+A

b
a
tanh(µt) tanh[βy(t)] dt = g(x).
This is a special case of equation 6.8.27 with f(t, y)=A tanh(µt) tanh(βy).
Page 395
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22. y(x)+A

b
a
e
λ(x–t)
tanh[βy(t)] dt = g(x).
This is a special case of equation 6.8.28 with f(t, y)=A tanh(βy).
23. y(x)+

b
a
g(x) tanh[βy(t)] dt = h(x).
This is a special case of equation 6.8.29 with f(t, y) = tanh(βy).
24. y(x)+A

b
a
cosh(λx + µt) tanh[βy(t)] dt = h(x).

This is a special case of equation 6.8.31 with f(t, y)=A tanh(βy).
25. y(x)+A

b
a
sinh(λx + µt) tanh[βy(t)] dt = h(x).
This is a special case of equation 6.8.32 with f(t, y)=A tanh(βy).
26. y(x)+A

b
a
cos(λx + µt) tanh[βy(t)] dt = h(x).
This is a special case of equation 6.8.33 with f(t, y)=A tanh(βy).
27. y(x)+A

b
a
sin(λx + µt) tanh[βy(t)] dt = h(x).
This is a special case of equation 6.8.34 with f(t, y)=A tanh(βy).
6.5-4. Integrands With Nonlinearity of the Form coth[βy(t)]
28. y(x)+A

b
a
coth[βy(t)] dt = g(x).
This is a special case of equation 6.8.27 with f(t, y)=A coth(βy).
29. y(x)+A

b
a

t
µ
coth
k
[βy(t)] dt = g(x).
This is a special case of equation 6.8.27 with f(t, y)=At
µ
coth
k
(βy).
30. y(x)+A

b
a
coth(µt) coth[βy(t)] dt = g(x).
This is a special case of equation 6.8.27 with f(t, y)=A coth(µt) coth(βy).
31. y(x)+A

b
a
e
λ(x–t)
coth[βy(t)] dt = g(x).
This is a special case of equation 6.8.28 with f(t, y)=A coth(βy).
32. y(x)+

b
a
g(x) coth[βy(t)] dt = h(x).
This is a special case of equation 6.8.29 with f(t, y) = coth(βy).

Page 396
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
33. y(x)+A

b
a
cosh(λx + µt) coth[βy(t)] dt = h(x).
This is a special case of equation 6.8.31 with f(t, y)=A coth(βy).
34. y(x)+A

b
a
sinh(λx + µt) coth[βy(t)] dt = h(x).
This is a special case of equation 6.8.32 with f(t, y)=A coth(βy).
35. y(x)+A

b
a
cos(λx + µt) coth[βy(t)] dt = h(x).
This is a special case of equation 6.8.33 with f(t, y)=A coth(βy).
36. y(x)+A

b
a
sin(λx + µt) coth[βy(t)] dt = h(x).
This is a special case of equation 6.8.34 with f(t, y)=A coth(βy).
6.5-5. Other Integrands
37. y(x)+A


b
a
cosh[βy(x)] cosh[γy(t)] dt = h(x).
This is a special case of equation 6.8.43 with g(x, y)=A cosh(βy) and f(t, y) = cosh(γy).
38. y(x)+A

b
a
y(xt) cosh[βy(t)] dt =0.
This is a special case of equation 6.8.45 with f(t, y)=A cosh(βy).
39. y(x)+A

b
a
sinh[βy(x)] sinh[γy(t)] dt = h(x).
This is a special case of equation 6.8.43 with g(x, y)=A sinh(βy) and f(t, y) = sinh(γy).
40. y(x)+A

b
a
y(xt) sinh[βy(t)] dt =0.
This is a special case of equation 6.8.45 with f(t, y)=A sinh(βy).
41. y(x)+A

b
a
tanh[βy(x)] tanh[γy(t)] dt = h(x).
This is a special case of equation 6.8.43 with g(x, y)=A tanh(βy) and f(t, y) = tanh(γy).
42. y(x)+A


b
a
y(xt) tanh[βy(t)] dt =0.
This is a special case of equation 6.8.45 with f(t, y)=A tanh(βy).
43. y(x)+A

b
a
coth[βy(x)] coth[γy(t)] dt = h(x).
This is a special case of equation 6.8.43 with g(x, y)=A coth(βy) and f(t, y) = coth(γy).
44. y(x)+A

b
a
y(xt) coth[βy(t)] dt =0.
This is a special case of equation 6.8.45 with f(t, y)=A coth(βy).
Page 397
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC