Handbook of mathematics for engineers and scienteists part 71 pot

The left-hand side of the equation is the total differential of a function of two variables U x, y.. An integrating factor for the equation f x, y dx + gx, y dy =0 is a function μx, y 0

12.1.2-6 Bernoulli equation y x  + f (x)y = g(x)y a.

A Bernoulli equation has the form

y 

x + f (x)y = g(x)y a, a≠ 0, 1

(For a = 0 and a = 1, it is a linear equation; see Paragraph 12.1.2-5.) The substitution

z = y1–a brings it to a linear equation, z x  + (1– a)f (x)z = (1– a)g(x), which is discussed

in Paragraph 12.1.2-5 With this in view, one can obtain the general integral:

y1 –a = Ce–F + (1– a)e–F



e F g (x) dx, where F = (1– a)



f (x) dx.

12.1.2-7 Equation of the form xy x  = y + f (x)g(y/x).

The substitution u = y/x brings the equation to a separable equation, x2u 

x = f (x)g(u);

see Paragraph 12.1.2-1

12.1.2-8 Darboux equation

A Darboux equation can be represented as

*

f

y

x



+ x a h

y

x

+

y 

x = g

y

x



+ yx a–1h

y

x



Using the substitution y = xz(x) and taking z to be the independent variable, one obtains

a Bernoulli equation, which is considered in Paragraph 12.1.2-6:



g (z) – zf (z)

x 

z = xf (z) + x a+1h (z).

 Some other first-order equations integrable by quadrature are treated in Section T5.1

12.1.3 Exact Differential Equations Integrating Factor

12.1.3-1 Exact differential equations

An exact differential equation has the form

f (x, y) dx + g(x, y) dy =0, where ∂f

∂y = ∂g

∂x The left-hand side of the equation is the total differential of a function of two variables

U (x, y).

The general integral, U (x, y) = C, where C is an arbitrary constant and the function U

is determined from the system:

∂U

∂y = g.

Integrating the first equation yields U =7

f (x, y) dx+ Ψ(y) (while integrating, the variable y

is treated as a parameter) On substituting this expression into the second equation, one identifies the function Ψ (and hence, U) As a result, the general integral of an exact

differential equation can be represented in the form

 x

x0f (ξ, y) dξ +  y

y0 g (x0, η) dη = C, where x0 and y0 are any numbers

TABLE 12.1

An integrating factor μ = μ(x, y) for some types of ordinary differential equations f dx + g dy =0 , where

f = f (x, y) and g = g(x, y) The subscripts x and y indicate the corresponding partial derivatives

No. Conditions for f and g Integrating factor Remarks

1 f = yϕ(xy), g = xψ(xy) μ= xf–yg1 xf – yg 0 ;

ϕ (z) and ψ(z) are any functions

2 f x = g y , f y = –g x μ= f21+g2 f + ig is an analytic function

of the complex variable x + iy

3 f y–g g x = ϕ(x) μ= exp  7

ϕ (x) dx

ϕ (x) is any function

4 f y–f g x = ϕ(y) μ= exp 

– 7

ϕ (y) dy

ϕ (y) is any function

5 f y g–f–g x = ϕ(x + y) μ= exp 7

ϕ (z) dz

, z = x + y ϕ (z) is any function

6 yg–xf f y–g x = ϕ(xy) μ= exp 7

ϕ (z) dz

, z = xy ϕ (z) is any function

7 x2yg+xf(f y–g x) = ϕ y x

μ= exp 

– 7

ϕ (z) dz

, z = x y ϕ (z) is any function

8 xg–yf f y–g x = ϕ(x2+ y2) μ= exp 1

2

7

ϕ (z) dz

, z = x2+y2 ϕ (z) is any function

9 f y – g x = ϕ(x)g – ψ(y)f μ= exp  7

ϕ (x) dx +7

ψ (y) dy

ϕ (x) and ψ(y) are any functions

10 gω f y x––g fω x y = ϕ(ω) μ= exp  7

ϕ (ω) dω ω = ω(x, y) is any function

of two variables

Example Consider the equation

(ay n + bx)y x  + by + cx m= 0 , or (by + cx m ) dx + (ay n + bx) dy =0 ,

defined by the functions f (x, y) = by + cx m and g(x, y) = ay n + bx Computing the derivatives, we have

∂f

∂y = b, ∂g

∂x = b =⇒ ∂f

∂y = ∂g

∂x Hence the given equation is an exact differential equation Its solution can be found using the last formula from

Paragraph 12.1.3-1 with x0= y0= 0 :

a

n+ 1y n+1+ bxy +

c

m+ 1x m+1= C.

12.1.3-2 Integrating factor

An integrating factor for the equation

f (x, y) dx + g(x, y) dy =0

is a function μ(x, y) 0 such that the left-hand side of the equation, when multiplied by

μ (x, y), becomes a total differential, and the equation itself becomes an exact differential

equation

An integrating factor satisfies the first-order partial differential equation,

g ∂μ

∂x – f ∂μ

∂y =



∂f

∂y – ∂g

∂x



μ, which is not generally easier to solve than the original equation

Table 12.1 lists some special cases where an integrating factor can be found in explicit form

12.1.4 Riccati Equation

12.1.4-1 General Riccati equation Simplest integrable cases

A Riccati equation has the general form

y 

x = f2(x)y2+ f1(x)y + f0(x). (12.1.4.1)

If f2 ≡ 0, we have a linear equation (see Paragraph 12.1.2-5), and if f0 ≡ 0, we have a

Bernoulli equation (see Paragraph 12.1.2-6 for a =2), whose solutions were given previously

For arbitrary f2, f1, and f0, the Riccati equation is not integrable by quadrature

Listed below are some special cases where the Riccati equation (12.1.4.1) is integrable

by quadrature

1◦ The functions f2, f1, and f0 are proportional, i.e.,

y 

x = ϕ(x)(ay2+ by + c), where a, b, and c are constants. This equation is a separable equation; see Para-graph 12.1.2-1

2◦ The Riccati equation is homogeneous:

y 

x = a y

2

x2 + b

y

x + c.

See Paragraph 12.1.2-3

3◦ The Riccati equation is generalized homogeneous:

y 

x = ax n y2+ x b y + cx–n–2. See Paragraph 12.1.2-4 (with k = –n–1) The substitution z = x n+1y brings it to a separable

equation: xz x  = az2+ (b + n +1)z + c.

4◦ The Riccati equation has the form

y 

x = ax2n y2+ m x – n y + cx2m.

By the substitution y = x m–n z , the equation is reduced to a separable equation: x–n–m z 

x=

az2+ c.

 Some other Riccati equations integrable by quadrature are treated in Section T5.1 (see equations T5.1.6 to T5.1.22)

12.1.4-2 Polynomial solutions of the Riccati equation

Let f2=1, f1(x), and f0(x) be polynomials If the degree of the polynomial

Δ = f12–2(f1) x–4f0

is odd, the Riccati equation cannot possess a polynomial solution If the degree of Δ is even, the equation involved may possess only the following polynomial solutions:

y= –12 f1 √

Δ, where√

Δ denotes an integer rational part of the expansion of√

Δ in decreasing powers

of x (for example, √

x2–2x+3= x –1)

12.1.4-3 Use of particular solutions to construct the general solution.

1◦ Given a particular solution y0 = y0(x) of the Riccati equation (12.1.4.1), the general

solution can be written as

y = y0(x) + Φ(x)*C–



Φ(x)f2(x) dx

+–1 , (12.1.4.2)

where C is an arbitrary constant and

Φ(x) = exp 2f2(x)y0(x) + f1(x)

dx

4 (12.1.4.3)

To the particular solution y0(x) there corresponds C = ∞.

2◦ Let y1= y1(x) and y2= y2(x) be two different particular solutions of equation (12.1.4.1).

Then the general solution can be calculated by

y= Cy1+ U (x)y2

C + U (x) , where U (x) = exp

*

f2(y1– y2) dx

+

To the particular solution y1(x), there corresponds C = ∞; and to y2 (x), there corresponds

C=0

3◦ Let y1 = y1(x), y2 = y2(x), and y3 = y3(x) be three distinct particular solutions of

equation (12.1.4.1) Then the general solution can be found without quadrature:

y – y2

y – y1

y3– y1

y3– y2 = C.

This means that the Riccati equation has a fundamental system of solutions

12.1.4-4 Some transformations

1◦ The transformation (ϕ, ψ1, ψ2, ψ3, and ψ4are arbitrary functions)

x = ϕ(ξ), y= ψ4(ξ)u + ψ3(ξ)

ψ2(ξ)u + ψ1(ξ)

reduces the Riccati equation (12.1.4.1) to a Riccati equation for u = u(ξ).

2◦ Let y0 = y0(x) be a particular solution of equation (12.1.4.1) Then the substitution

y = y0+1/w leads to a linear equation for w = w(x):

w 

x+

2f2(x)y0(x) + f1(x)

w + f2(x) =0 For solution of linear equations, see Paragraph 12.1.2-5

12.1.4-5 Reduction of the Riccati equation to a second-order linear equation

The substitution

u (x) = exp

 –



f2y dx

 reduces the general Riccati equation (12.1.4.1) to a second-order linear equation:

f2u 

xx–



(f2) x + f1f2

u 

x + f0f2

2u=0, which often may be easier to solve than the original Riccati equation

12.1.4-6 Reduction of the Riccati equation to the canonical form.

The general Riccati equation (12.1.4.1) can be reduced with the aid of the transformation

x = ϕ(ξ), y= 1

F2w–

1 2

F1

F2 +

1 2

 1

F2



ξ

, where F i (ξ) = f i (ϕ)ϕ  ξ, (12.1.4.4)

to the canonical form

w 

ξ = w2+Ψ(ξ). (12.1.4.5) Here, the function Ψ is defined by the formula

Ψ(ξ) = F0F2– 1

4F12+ 1

2F1 – 1

2F1

F 

2

F2 –

3 4



F 

2

F2

2 + 1 2

F 

2

F2;

the prime denotes differentiation with respect to ξ.

Transformation (12.1.4.4) depends on a function ϕ = ϕ(ξ) that can be arbitrary For a specific original Riccati equation, different functions ϕ in (12.1.4.4) will generate different

functions Ψ in equation (12.1.4.5) In practice, transformation (12.1.4.4) is most frequently

used with ϕ(ξ) = ξ.

12.1.5 Abel Equations of the First Kind

12.1.5-1 General form of Abel equations of the first kind Some integrable cases

An Abel equation of the first kind has the general form

y 

x = f3(x)y3+ f2(x)y2+ f1(x)y + f0(x), f3(x)0 (12.1.5.1)

In the degenerate case f2(x) = f0(x) = 0, we have a Bernoulli equation (see

Para-graph 12.1.2-6 with a =3) The Abel equation (12.1.5.1) is not integrable in closed form

for arbitrary f n (x).

Listed below are some special cases where the Abel equation of the first kind is integrable

by quadrature

1◦ If the functions f

n (x) (n = 0, 1, 2,3) are proportional, i.e., f n (x) = a n g (x), then

(12.1.5.1) is a separable equation (see Paragraph 12.1.2-1)

2◦ The Abel equation is homogeneous:

y 

x = a y

3

x3 + b

y2

x2 + c

y

x + d.

See Paragraph 12.1.2-3

3◦ The Abel equation is generalized homogeneous:

y 

x = ax2n+1y3+ bx n y2+ x c y + dx–n–2. See Paragraph 12.1.2-4 for k = –n –1 The substitution w = x n+1y leads to a separable

equation: xw x  = aw3+ bw2+ (c + n +1)w + d.

4◦ The Abel equation

y 

x = ax3n–m y3+ bx2n y2+ m x – n y + dx2m can be reduced with the substitution y = x m–n z to a separable equation: x–n–m z 

x =

az3+ bz2+ c.

5◦ Let f0 ≡ 0, f1 ≡ 0, and (f3/f2) x = af2 for some constant a Then the substitution

y = f2f–1

3 u leads to a separable equation: u  x = f22f3–1(u3+ u2+ au).

6◦ If

f0= f31f f32 –

2f3 2

27f2 3

– 1 3

d dx

f2

f3, f n = f n (x),

then the solution of equation (12.1.5.1) is given by

y (x) = E



C–2 f3E2dx–1 2

– f2

3f3, where E= exp

 

f1– f

2 2

3f3



dx



For other solvable Abel equations of the first kind, see the books by Kamke (1977) and Polyanin and Zaitsev (2003)

12.1.5-2 Reduction of the Abel equation of the first kind to the canonical form The transformation

y = U (x)η(ξ) – f2

3f3, ξ=



f3U2dx, where U (x) = exp

f1– f

2 2

3f3



dx

 , brings equation (12.1.5.1) to the canonical (normal) form

η 

ξ = η3+Φ(ξ).

Here, the function Φ(ξ) is defined parametrically (x is the parameter) by the relations

Φ = 1

f3U3



f0– f31f f32 +

2f3 2

27f2 3

+ 1 3

d dx

f2

f3

 , ξ= f3U2dx.

12.1.5-3 Reduction to an Abel equation of the second kind

Let y0= y0(x) be a particular solution of equation (12.1.5.1) Then the substitution

y = y0+ E (x)

z (x) , where E (x) = exp

*

(3f3y2

0+2f2y0+ f1) dx

+ , leads to an Abel equation of the second kind:

zz 

x= –(3f3y0+ f2)Ez – f3E2.

For equations of this type, see Subsection 12.1.6

12.1.6 Abel Equations of the Second Kind

12.1.6-1 General form of Abel equations of the second kind Some integrable cases

An Abel equation of the second kind has the general form

[y + g(x)]y x  = f2(x)y2+ f1(x)y + f0(x), g (x)0 (12.1.6.1)

The Abel equation (12.1.6.1) is not integrable for arbitrary f n (x) and g(x) Given

below are some special cases where the Abel equation of the second kind is integrable by quadrature

1◦ If g(x) = const and the functions f

n (x) (n = 0,1, 2) are proportional, i.e., f n (x) =

a n g (x), then (12.1.6.1) is a separable equation (see Paragraph 12.1.2-1).

2◦ The Abel equation is homogeneous:

(y + sx)y  x= a

x y

2+ by + cx.

See Paragraph 12.1.2-3 The substitution w = y/x leads to a separable equation.

3◦ The Abel equation is generalized homogeneous:

(y + sx n )y  x= a

x y

2+ bx n–1y + cx2n–1.

See Paragraph 12.1.2-4 for k = n The substitution w = yx–nleads to a separable equation:

x (w + s)w  x = (a – n)w2+ (b – ns)w + c.

4◦ The Abel equation

(y + a2x + c2)y x  = b1y + a1x + c1

is a special case of the equation treated in Paragraph 12.1.2-3 (see Item2◦ with f (w) = w and b2=1)

5◦ The unnormalized Abel equation

[(a1x + a2x n )y + b1x + b2x n ]y 

x = c2y2+ c1y + c0 can be reduced to the form (12.1.6.1) by dividing it by (a1x + a2x n ) Taking y to be the independent variable and x = x(y) to be the dependent one, we obtain the Bernoulli equation

(c2y2+ c

1y + c0)x  y = (a1y + b1)x + (a2y + b2)x n See Paragraph 12.1.2-6

6◦ The general solution of the Abel equation

(y + g)y  x = f2y2+ f

1y + f1g – f2g2, f

n = f n (x), g = g(x),

is given by

y = –g + CE + E(f1+ g  x–2f2g )E–1dx, where E= exp



f2dx



7◦ If f1 =2f2g – g x , the general solution of the Abel equation (12.1.6.1) has the form

y = –g E

*

2(f0+ gg x  – f2g2)E– 2dx + C+1 2

, where E = exp

f2dx

 For other solvable Abel equations of the second kind, see the books by Kamke (1977) and Polyanin and Zaitsev (2003)