HANDBOOK OFINTEGRAL EQUATIONS phần 6 docx

For sufficiently largeλ, the integral equation has some other more complicated solutions of the polynomial form yx =... The equation has more complicated solutions of the formyx = e–βxn

6.1-2 Equations of the Formb

(2C + 1)x C–C – 1

(2C + 1)x C–C – 1

,

y10(x) = – √

A (ln x + 1),

whereC is an arbitrary constant.

3◦ See 6.2.2 for some other solutions

,

whereB =



2A β(β + 1).

2◦ The integral equation has some other (more complicated solutions) of the polynomial

A(2λ – 1).

2◦ For sufficiently largeλ, the integral equation has some other (more complicated) solutions

of the polynomial form y(x) =

B k x k, where the constantsB k can be found from the corresponding system

of algebraic equations See 6.2.2 for some other solutions

6.1-3 Equations of the Formy(x) +b

This is a special case of equation 6.2.22 withg(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

This is a special case of equation 6.2.23 withg(x) = Ax βandf (x) = Bx µ.

30. y(x) + A

 b

a

e βx y2 (t) dt = Be µx.

This is a special case of equation 6.2.23 withg(x) = Ae βxandf (x) = Be µx

6.1-4 Equations of the Formy(x) +b

This is a special case of equation 6.2.26 withg(x) = Ae βxandf (x) = Be µx

6.1-5 Equations of the Formy(x) +b

whereC is an arbitrary nonnegative constant.

There are more complicated solutions of the formy(x) = x Cn

k=0

B k x k, whereC is an

arbitrary constant and the coefficients B k can be found from the corresponding system of

algebraic equations

2◦ A solution:

y3(x) = (I1–I0)x

β+I1–I2I0I2–I2 x C, I m= A

2C + mβ + 1, m = 0, 1, 2,

whereC and β are arbitrary constants.

There are more complicated solutions of the formy(x) = x Cn

t2C(lnt) m dt, m = 0, 1, 2,

whereC is an arbitrary constant.

There are more complicated solutions of the formy(x) = x C

Cx

,whereC is an arbitrary constant.

2◦ There are more complicated solutions of the formy(x) = e Cx

6.2 Equations With Quadratic Nonlinearity That Contain

whereC is an arbitrary constant.

The equation has more complicated solutions of the formy(x) =

* The arguments of the equations containingy(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)

iff (t) is nonzero only on the interval a ≤ t ≤ b.

2◦ The substitutiony(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)x2β.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 formy(x) = p ln x + q The constants p and q are determined

from the following system of two second-order algebraic equations:

I1p2+I0pq = A, I2p2+ 2I1pq + I0q2=B,

where

I m=

 b a

I m=

 b a

t m f (t) dt, q =



A λ(I2

A solution: y(x) = βx + µ, where the constants β and µ are determined from the following

system of two second-order algebraic equations:

I0βµ + I1β2=A, I0µ2+ (λ + 1)I1βµ + λI2β2=B, I m=

 b a

t m f (t) dt. (1)Multiplying the first equation byB and the second by –A and adding the resulting equations,

we obtain the quadratic equation

AI0z2+

(λ + 1)AI1–BI0

z + λAI2–BI1= 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

2◦ The equation has more complicated solutions of the formy(x) = e–βxn

k=0

B k x k, wherethe constantsB kcan be found from the corresponding system of algebraic equations

* The arguments of the equations containingy(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) iff (t) is nonzero only on the interval a ≤ t ≤ b.

3◦ The substitutiony(x) = e–βx w(x) leads to an equation of the form 6.2.7:

I m=

 b a

λ k x k, where the constantsλ kcan be found from the corresponding system

of algebraic equations Forn = 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:

I0λµ + I1λ2=A, I0µ2–I2λ2=B, I m=

 b a

t m f (t) dt, m = 0, 1, 2. (1)Multiplying the first equation byB and the second by –A and adding the results, we obtain

the quadratic equation

AI0z2–BI0z – AI2–BI1= 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



k=0

where the constantsλ 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 ofx.

* The arguments of the equations containingy(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) iff (t) is nonzero only on the interval a ≤ t ≤ b.

The integral equation has more complicated solutions of the formy(x) = e λxn

k=0

B k x k, wherethe constantsB kcan be found from the corresponding system of algebraic equations

Herep and q are roots of the algebraic system

I0pq + Ics(p2–q2) =A, Iccq2–Issp2= 0, (2)where the notation

I0=

 b a

f (t) dt, Ics=

 b a

It follows from the second equation of (2) thatq = ± Iss/Icc p Using this expression to

eliminateq from the first equation of (2), we obtain the following four solutions:

Herep and q are roots of the algebraic system

I0pq + Ics(p2–q2) = 0, Icc q2–Issp2=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

Herep and q are roots of the algebraic system

I0pq + Ics(p2+q2) =A, Iccq2–Issp2= 0, (2)where

 b a

f (t) cos2(λt) dt, Iss=

 b a

f (t) sin2(λt) dt.

It follows from the second equation of (2) thatq = ± Iss/Icc p Using this expression to

eliminateq from the first equation of (2), we obtain the following four solutions:

Herep and q are roots of the algebraic system

I0pq + Ics(p2+q2) = 0, Iccq2–Issp2=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

6.2-2 Equations of the Formy(x) +b

This is a special case of equation 6.8.29

Solutions: y1(x) = 0 and y2(x) = λf (x), where λ = –

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)λ2+ (1 + 2AI1)λ + AI2= 0, where I1=

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

I0λ2+ (1 + 2I1)λ + I2= 0, where I m=

 b a

g2(t) dt, I f g=

 b a

f (t)g(t) dt, I f f =

 b a

A solution: y(x) = λ1g1(x) + λ2g2(x) + f (x), where the constants λ1 andλ2 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)

6.2-3 Equations of the Formy(x) +b

Here the function y(x) = y(x, λ) obtained by solving the quadratic equation (1) must be

substituted in the integrand of (2)

Here the function y(x) = y(x, λ) obtained by solving the quadratic equation (1) must be

substituted into the integrand of (2)

6.2-4 Equations of the Formy(x) +b

f (t)t2C+m dt, m = 0, 1, 2,

whereC is an arbitrary constant.

There are more complicated solutions of the formy(x) = x Cn

whereC and β are arbitrary constants.

There are more complicated solutions of the formy(x) = x Cn

whereC is an arbitrary constant.

There are more complicated solutions of the formy(x) = x C

4◦ The equation also has the trivial solutiony(x)≡ 0

5◦ The substitutiony(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)x2β

whereλ and µ are determined from the following system of two algebraic equations (this

system can be reduced to a quadratic equation):

This equation has solutions of the formy(x) = p ln x + q, where the constants p and q can be

found from a system of two second-order algebraic equations

2◦ The equation has the trivial solutiony(x)≡ 0

3◦ The substitutiony(x) = x β w(x) leads to an equation of the same form,

Iλ2+λ – A = 0, I =

 ∞

0

f (t) dt.

t mexp–C(λ + 1)t

f (t) dt, m = 0, 1, 2,

whereC is an arbitrary constant.

2◦ There are more complicated solutions of the formy(x) = exp(–Cx)

3◦ The equation also has the trivial solutiony(x)≡ 0

4◦ The substitutiony(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).

can be found from the corresponding system of algebraic equations

3◦ The substitutiony(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

t m f (t) dt,

whereC is an arbitrary constant and m = 0, 1, 2.

2◦ There are more complicated solutions of the formy(x) = exp(Cx)

3◦ The equation also has the trivial solutiony(x)≡ 0

4◦ The substitutiony(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.

Ik2+k – A = 0, I =

 b a

f (t) dt.

2◦ The substitutiony(x) = e βx w(x) leads to an equation of the same form,

w(x) +

 b a

Herep and q are roots of the algebraic system

p + I0pq + Ics(p2–q2) =A, q + Iccq2–Issp2= 0, (2)where

I0=

 b a

f (t) dt, Ics=

 b a

Herep and q are roots of the algebraic system

p + I0pq + Ics(p2–q2) = 0, q + Iccq2–Issp2=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

Herep and q are roots of the algebraic system

p + I0pq + Ics(p2+q2) =A, q + Iccq2–Issp2 = 0, (2)where

I0=

 b a

f (t) dt, Ics=

 b a

f (t) cos(λt) sin(λt) dt,

Icc=

 b a

f (t) cos2(λt) dt, Iss=

 b a

f (t) sin2(λt) dt.

Different solutions of system (2) generate different solutions (1) of the integral equation

Herep and q are roots of the algebraic system

p + I0pq + Ics(p2+q2) = 0, q + Iccq2–Issp2=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 Formb

g β(t) dt

 1 1–β

.Forβ > 0, the equation also has the trivial solution y(x)≡ 0

aβ + b x

, A1–β=

6.3-3 Equations of the Formy(x) +b

Here the function y(x) = y(x, λ) obtained by solving the quadratic equation (1) must be

substituted in the integrand of (2)

g1(t) dt, (1)whereλ is determined by the algebraic (or transcendental) equation

λ =

 b

a

Here the function y(x) = y(x, λ) obtained by solving the quadratic equation (1) must be

substituted in the integrand of (2)

6.4 Equations With Exponential Nonlinearity

6.4-1 Integrands With Nonlinearity of the Form exp[βy(t)]

6.5 Equations With Hyperbolic Nonlinearity

6.5-1 Integrands With Nonlinearity of the Form cosh[βy(t)]

This is a special case of equation 6.8.34 withf (t, y) = A cosh(βy).

6.5-2 Integrands With Nonlinearity of the Form sinh[βy(t)]

This is a special case of equation 6.8.34 withf (t, y) = A sinh(βy).

6.5-3 Integrands With Nonlinearity of the Form tanh[βy(t)]

This is a special case of equation 6.8.34 withf (t, y) = A tanh(βy).

6.5-4 Integrands With Nonlinearity of the Form coth[βy(t)]

cosh[βy(x)] cosh[γy(t)] dt = h(x).

This is a special case of equation 6.8.43 withg(x, y) = A cosh(βy) and f (t, y) = cosh(γy).

sinh[βy(x)] sinh[γy(t)] dt = h(x).

This is a special case of equation 6.8.43 withg(x, y) = A sinh(βy) and f (t, y) = sinh(γy).

tanh[βy(x)] tanh[γy(t)] dt = h(x).

This is a special case of equation 6.8.43 withg(x, y) = A tanh(βy) and f (t, y) = tanh(γy).

coth[βy(x)] coth[γy(t)] dt = h(x).

This is a special case of equation 6.8.43 withg(x, y) = A coth(βy) and f (t, y) = coth(γy).

6.6 Equations With Logarithmic Nonlinearity

6.6-1 Integrands With Nonlinearity of the Form ln[βy(t)]

6.7 Equations With Trigonometric Nonlinearity

6.7-1 Integrands With Nonlinearity of the Form cos[βy(t)]

This is a special case of equation 6.8.34 withf (t, y) = A cos(βy).

6.7-2 Integrands With Nonlinearity of the Form sin[βy(t)]

This is a special case of equation 6.8.34 withf (t, y) = A sin(βy).

6.7-3 Integrands With Nonlinearity of the Form tan[βy(t)]

This is a special case of equation 6.8.34 withf (t, y) = A tan(βy).

6.7-4 Integrands With Nonlinearity of the Form cot[βy(t)]

cos[βy(x)] cos[γy(t)] dt = h(x).

This is a special case of equation 6.8.43 withg(x, y) = A cos(βy) and f (t, y) = cos(γy).

sin[βy(x)] sin[γy(t)] dt = h(x).

This is a special case of equation 6.8.43 withg(x, y) = A sin(βy) and f (t, y) = sin(γy).

tan[βy(x)] tan[γy(t)] dt = h(x).

This is a special case of equation 6.8.43 withg(x, y) = A tan(βy) and f (t, y) = tan(γy).

cot[βy(x)] cot[γy(t)] dt = h(x).

This is a special case of equation 6.8.43 withg(x, y) = A cot(βy) and f (t, y) = cot(γy).

6.8 Equations With Nonlinearity of General Form

6.8-1 Equations of the Formb

f

t, y(t)

Here the functiony(x) = y(x, λ) obtained by solving (1) must be substituted into (2).

The number of solutions of the integral equation is determined by the number of the

solutions obtained from (1) and (2)

2◦ Solutions: y(x) = px + q, where p and q are roots of the following system of algebraic

(or transcendental) equations:

= ¯f (t)y(t), see 6.2.2 for solutions of this system.

2◦ The integral equation has some other (more complicated) solutions of the polynomial

formy(x) =

n



k=0

B k x k, where the constantsB k can be found from the corresponding system

of algebraic (or transcendental) equations

4◦ The integral equation can have logarithmic solutions similar to those presented in item 3◦

of equation 6.2.2

tf (t, pt + q) dt – A = 0, q

 b a

f (t, pt + q) dt – B = 0. (2)

Different solutions of system (2) generate different solutions (1) of the integral equation

2◦ The integral equation has some other (more complicated) solutions of the polynomial

formy(x) =

n



k=0

B k x k, where the constantsB k can be found from the corresponding system

of algebraic (or transcendental) equations

This equation has solutions of the formy(x) = px βcos(β ln x) + qx βsin(β ln x), where p and q

are some constants

f (t, pt + q) dt – A = 0,

 b a

(βpt + q)f (t, pt + q) dt – B = 0. (2)Different solutions of system (2) generate different solutions (1) of the integral equation

This is a special case of equation 6.8.35 withf (t, y) = f (y) and g(x) = Ax2+Bx + C.

The functiony = y(x) obeys the second-order autonomous differential equation

f (t) dt, (1)

wherey a =y(a) and w a = y  x(a) are constants of integration These constants, as well as

the unknownsy b =y(b) and w b =y  x(b), are determined by the algebraic (or transcendental)

Here the first equation is obtained from the second condition of (5) in 6.8.35, the second

equation is obtained from condition (6) in 6.8.35, and the third and fourth equations are

This is a special case of equation 6.8.36 withf (t, y) = f (y) and g(x) = A + Be λx+Ce–λx

The functiony = y(x) satisfies the second-order autonomous differential equation

y xx+ 2λf (y) – λ2y = –λ2A, (1)

whose solution can be written in an implicit form:

f (t) dt, (2)

wherey a =y(a) and w a = y  x(a) are constants of integration These constants, as well as

the unknownsy b =y(b) and w b =y  x(b), are determined by the algebraic (or transcendental)

system

w a+λy a=Aλ + 2Bλe λa,

w b–λy b = –Aλ – 2Cλe–λb,

Here the first and second equations are obtained from conditions (5) in 6.8.86, and the third

and fourth equations are consequences of (2)

Each solution of system (3) generates a solution of the integral equation

f (t) dt,

wherey a=y(a) and w a =y  x(a) are constants of integration, which can be determined from

the boundary conditions (5) in 6.8.37

This is a special case of equation 6.8.38 withf (t, y) = f (y) and g(x) = A+B cos(λx)+C sin(λx).

The functiony = y(x) satisfies the second-order autonomous differential equation

wherey a=y(a) and w a =y  x(a) are constants of integration, which can be determined from

the boundary conditions (5) in 6.8.38

6.8-3 Equations of the Formy(x) +b

A solution: y(x) = g(x) + λx + µ, where the constants λ and µ are determined from the

algebraic (or transcendental) system

λ + A

 b a

f

t, g(t) + λt + µ

dt = 0, µ + B

 b a

Using the formula cosh(λx + µt) = cosh(λx) cosh(µt) + sinh(µt) sinh(λx), we arrive at an

equation of the form 6.8.39:

y(x) +

 b a

cosh(λx)f1

t, y(t)+ sinh(λx)f2

t, y(t)

= sinh(µt)f

t, y(t)

Using the formula sinh(λx + µt) = cosh(λx) sinh(µt) + cosh(µt) sinh(λx), we arrive at an

equation of the form 6.8.39:

y(x) +

 b a

cosh(λx)f1

t, y(t)+ sinh(λx)f2

t, y(t)

dt = h(x), f1

t, y(t)

= sinh(µt)f

t, y(t), f2

t, y(t)

= cosh(µt)f

t, y(t)

t, y(t)+ sin(λx)f2

t, y(t)

dt = h(x), f1

t, y(t)

= cos(µt)f

t, y(t), f2

t, y(t)

= – sin(µt)f

t, y(t)

cos(λx)f1

t, y(t)+ sin(λx)f2

t, y(t)

dt = h(x), f1

t, y(t)

= sin(µt)f

t, y(t), f2

t, y(t)

= cos(µt)f

t, y(t)

(x – t)f

t, y(t)

dt +

 b x

f

t, y(t)

dt –

 b x

f

t, y(t)

dt = g x (x). (2)Differentiating (2), we arrive at a second-order ordinary differential equation fory = y(x):

y xx+ 2f (x, y) = g xx  (x). (3)

2◦ Let us derive the boundary conditions for equation (3) We assume that –∞ < a < b < ∞.

By settingx = a and x = b in (1), we obtain the relations

y(a) +

 b a

(t – a)f

t, y(t)

dt = g(a), y(b) +

Let us solve equation (3) forf (x, y) and substitute the result into (4) Integrating by parts

yields the desired boundary conditions fory(x):

y(a) + y(b) + (b – a)

g  x(b) – y  x(b)

=g(a) + g(b), y(a) + y(b) + (a – b)

g  x(a) – y x (a)

Let us point out a useful consequence of (5):

y  x(a) + y  x(b) = g x (a) + g  x(b), (6)which can be used together with one of conditions (5)

Equation (3) under the boundary conditions (5) determines the solution of the original

integral equation (there may be several solutions) Conditions (5) make it possible to calculate

the constants of integration that occur in solving the differential equation (3)

Eliminating the integral terms from (1) and (2), we arrive at a second-order ordinary

differential equation fory = y(x):

y xx+ 2λf (x, y) – λ2y = g xx  (x) – λ2g(x). (3)

2◦ Let us derive the boundary conditions for equation (3) We assume that –∞ < a < b < ∞.

By settingx = a and x = b in (1), we obtain the relations

 b a

e λb ϕ  x(b) – e λa ϕ  x(a) = λe λa ϕ(a) + λe λb ϕ(b), ϕ(x) = y(x) – g(x);

e–λb ϕ  x(b) – e–λa ϕ  x(a) = λe–λa ϕ(a) + λe–λb ϕ(b).

Hence, we obtain the boundary conditions fory(x):

ϕ  x(a) + λϕ(a) = 0, ϕ  x(b) – λϕ(b) = 0; ϕ(x) = y(x) – g(x). (5)Equation (3) under the boundary conditions (5) determines the solution of the original

integral equation (there may be several solutions) Conditions (5) make it possible to calculate

the constants of integration that occur in solving the differential equation (3)

sinh[λ(x – t)]f

t, y(t)

dt +

 b x

sinh[λ(t – x)]f

t, y(t)

dt = g(x). (1)Differentiating (1) with respect tox twice yields

y xx(x) + 2λf

x, y(x)

+λ2

 x a

Eliminating the integral terms from (1) and (2), we arrive at a second-order ordinary

differential equation fory = y(x):

y xx+ 2λf (x, y) – λ2y = g xx  (x) – λ2g(x). (3)

2◦ Let us derive the boundary conditions for equation (3) We assume that –∞ < a < b < ∞.

By settingx = a and x = b in (1), we obtain the relations

y(a) +

 b a

sinh[λ(t – a)]f

t, y(t)

dt = g(a), y(b) +

sinh[λ(b – a)]ϕ  x(b) – λ cosh[λ(b – a)]ϕ(b) = λϕ(a), ϕ(x) = y(x) – g(x);

sinh[λ(b – a)]ϕ  x(a) + λ cosh[λ(b – a)]ϕ(a) = –λϕ(b). (5)

Equation (3) under the boundary conditions (5) determines the solution of the original

integral equation (there may be several solutions) Conditions (5) make it possible to calculate

the constants of integration that occur in solving the differential equation (3)

y xx(x) + 2λf

x, y(x)

–λ2

 x a

differential equation fory = y(x):

y xx+ 2λf (x, y) + λ2y = g xx  (x) + λ2g(x). (3)

2◦ Let us derive the boundary conditions for equation (3) We assume that –∞ < a < b < ∞.

By settingx = a and x = b in (1), we obtain the relations

 b a

sin[λ(b – a)] ϕ  x(b) – λ cos[λ(b – a)] ϕ(b) = λϕ(a), ϕ(x) = y(x) – g(x);

sin[λ(b – a)] ϕ  x(a) + λ cos[λ(b – a)] ϕ(a) = –λϕ(b). (5)

Equation (3) under the boundary conditions (5) determines the solution of the original

integral equation (there may be several solutions) Conditions (5) make it possible to calculate

the constants of integration that occur in solving the differential equation (3)

6. 6 Equations With Logarithmic Nonlinearity

6. 6-1 Integrands With Nonlinearity... condition of (5) in 6. 8.35, the second

equation is obtained from condition (6) in 6. 8.35, and the third and fourth equations are

This is a special case of equation 6. 8. 36 withf (t,...

the boundary conditions (5) in 6. 8.38