Handbook of mathematics for engineers and scienteists part 71 pot

458 ORDINARY DIFFERENTIAL EQUATIONS 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 = y 1–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: y 1–a = Ce –F +(1 – a)e –F  e F g(x) dx,whereF =(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, x 2 u  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–1 h  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+1 h(z).  Some other fi rst-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,whereC is an arbitrary constant and the function U is determined from the system: ∂U ∂x = f , ∂U ∂y = g. Integrating the first equation yields U =  f(x, y) dx+Ψ(y) (while integrating, thevariabley 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 x 0 f(ξ,y)dξ +  y y 0 g(x 0 , η) dη = C, where x 0 and y 0 are any numbers. 12.1. FIRST-ORDER DIFFERENTIAL EQUATIONS 459 TABLE 12.1 An integrating factor μ = μ(x, y) for some types of ordinary differential equations fdx+ gdy= 0,where f = f(x, y)andg = 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 = 1 xf–yg xf – yg 0; ϕ(z)andψ(z) are any functions 2 f x = g y , f y =–g x μ = 1 f 2 +g 2 f + ig is an analytic function of the complex variable x + iy 3 f y –g x g = ϕ(x) μ =exp   ϕ(x) dx  ϕ(x) is any function 4 f y –g x f = ϕ(y) μ =exp  –  ϕ(y) dy  ϕ(y) is any function 5 f y –g x g–f = ϕ(x + y) μ =exp   ϕ(z) dz  , z = x + y ϕ(z) is any function 6 f y –g x yg–xf = ϕ(xy) μ =exp   ϕ(z) dz  , z = xy ϕ(z) is any function 7 x 2 (f y –g x ) yg+xf = ϕ  y x  μ =exp  –  ϕ(z) dz  , z = y x ϕ(z) is any function 8 f y –g x xg–yf = ϕ(x 2 + y 2 ) μ =exp  1 2  ϕ(z) dz  , z = x 2 +y 2 ϕ(z) is any function 9 f y – g x = ϕ(x)g – ψ(y)f μ =exp   ϕ(x) dx +  ψ(y) dy  ϕ(x)andψ(y) are any functions 10 f y –g x gω x –fω y = ϕ(ω) μ =exp   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 x 0 = y 0 = 0: a n + 1 y n+1 + bxy + c m + 1 x 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. 460 ORDINARY DIFFERENTIAL EQUATIONS 12.1.4. Riccati Equation 12.1.4-1. General Riccati equation. Simplest integrable cases. A Riccati equation has the general form y  x = f 2 (x)y 2 + f 1 (x)y + f 0 (x). (12.1.4.1) If f 2 ≡ 0, we have a linear equation (see Paragraph 12.1.2-5), and if f 0 ≡ 0,wehavea Bernoulli equation(see Paragraph 12.1.2-6 fora=2), whose solutions were givenpreviously. For arbitrary f 2 , f 1 ,andf 0 , 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 f 2 , f 1 ,andf 0 are proportional, i.e., y  x = ϕ(x)(ay 2 + by + c), where a, b,andc 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 x 2 + b y x + c. See Paragraph 12.1.2-3. 3 ◦ . The Riccati equation is generalized homogeneous: y  x = ax n y 2 + b x y + cx –n–2 . See Paragraph 12.1.2-4 (with k =–n–1). The substitution z = x n+1 y brings it to a separable equation: xz  x = az 2 +(b + n + 1)z + c. 4 ◦ . The Riccati equation has the form y  x = ax 2n y 2 + m – n x y + cx 2m . By the substitution y = x m–n z, the equation is reduced to a separable equation: x –n–m z  x = az 2 + 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 f 2 = 1, f 1 (x), and f 0 (x) be polynomials. If the degree of the polynomial Δ = f 2 1 – 2(f 1 )  x – 4f 0 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 =– 1 2  f 1  √ Δ  , where  √ Δ  denotes an integer rational part of the expansion of √ Δ in decreasing powers of x (for example,  √ x 2 – 2x + 3  = x – 1). 12.1. FIRST-ORDER DIFFERENTIAL EQUATIONS 461 12.1.4-3. Use of particular solutions to construct the general solution. 1 ◦ . Given a particular solution y 0 = y 0 (x) of the Riccati equation (12.1.4.1), the general solution can be written as y = y 0 (x)+Φ(x)  C –  Φ(x)f 2 (x) dx  –1 ,(12.1.4.2) where C is an arbitrary constant and Φ(x)=exp    2f 2 (x)y 0 (x)+f 1 (x)  dx  .(12.1.4.3) To the particular solution y 0 (x) there corresponds C = ∞. 2 ◦ .Lety 1 = y 1 (x)andy 2 = y 2 (x) be two different particular solutions of equation (12.1.4.1). Then the general solution can be calculated by y = Cy 1 + U(x)y 2 C + U (x) ,whereU(x)=exp   f 2 (y 1 – y 2 ) dx  . To the particular solution y 1 (x), there corresponds C = ∞;andtoy 2 (x), there corresponds C = 0. 3 ◦ .Lety 1 = y 1 (x), y 2 = y 2 (x), and y 3 = y 3 (x) be three distinct particular solutions of equation (12.1.4.1). Then the general solution can be found without quadrature: y – y 2 y – y 1 y 3 – y 1 y 3 – y 2 = C. This means that the Riccati equation has a fundamental system of solutions. 12.1.4-4. Some transformations. 1 ◦ . The transformation (ϕ, ψ 1 , ψ 2 , ψ 3 ,andψ 4 are arbitrary functions) x = ϕ(ξ), y = ψ 4 (ξ)u + ψ 3 (ξ) ψ 2 (ξ)u + ψ 1 (ξ) reduces the Riccati equation (12.1.4.1) to a Riccati equation for u = u(ξ). 2 ◦ .Lety 0 = y 0 (x) be a particular solution of equation (12.1.4.1). Then the substitution y = y 0 + 1/w leads to a linear equation for w = w(x): w  x +  2f 2 (x)y 0 (x)+f 1 (x)  w + f 2 (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  –  f 2 ydx  reduces the general Riccati equation (12.1.4.1) to a second-order linear equation: f 2 u  xx –  (f 2 )  x + f 1 f 2  u  x + f 0 f 2 2 u = 0, which often may be easier to solve than the original Riccati equation. 462 ORDINARY DIFFERENTIAL EQUATIONS 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 F 2 w – 1 2 F 1 F 2 + 1 2  1 F 2   ξ ,whereF i (ξ)=f i (ϕ)ϕ  ξ ,(12.1.4.4) to the canonical form w  ξ = w 2 + Ψ(ξ). (12.1.4.5) Here, the function Ψ is defined by the formula Ψ(ξ)=F 0 F 2 – 1 4 F 2 1 + 1 2 F  1 – 1 2 F 1 F  2 F 2 – 3 4  F  2 F 2  2 + 1 2 F  2 F 2 ; 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 = f 3 (x)y 3 + f 2 (x)y 2 + f 1 (x)y + f 0 (x), f 3 (x) 0.(12.1.5.1) In the degenerate case f 2 (x)=f 0 (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 x 3 + b y 2 x 2 + c y x + d. See Paragraph 12.1.2-3. 3 ◦ . The Abel equation is generalized homogeneous: y  x = ax 2n+1 y 3 + bx n y 2 + c x y + dx –n–2 . See Paragraph 12.1.2-4 for k =–n – 1. The substitution w = x n+1 y leads to a separable equation: xw  x = aw 3 + bw 2 +(c + n + 1)w + d. 12.1. FIRST-ORDER DIFFERENTIAL EQUATIONS 463 4 ◦ . The Abel equation y  x = ax 3n–m y 3 + bx 2n y 2 + m – n x y + dx 2m can be reduced with the substitution y = x m–n z to a separable equation: x –n–m z  x = az 3 + bz 2 + c. 5 ◦ .Letf 0 ≡ 0, f 1 ≡ 0,and(f 3 /f 2 )  x = af 2 for some constant a. Then the substitution y = f 2 f –1 3 u leads to a separable equation: u  x = f 2 2 f –1 3 (u 3 + u 2 + au). 6 ◦ .If f 0 = f 1 f 2 3f 3 – 2f 3 2 27f 2 3 – 1 3 d dx f 2 f 3 , f n = f n (x), then the solution of equation (12.1.5.1) is given by y(x)=E  C – 2  f 3 E 2 dx  –1/2 – f 2 3f 3 ,whereE =exp    f 1 – f 2 2 3f 3  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)– f 2 3f 3 , ξ =  f 3 U 2 dx,whereU(x)=exp    f 1 – f 2 2 3f 3  dx  , brings equation (12.1.5.1) to the canonical (normal) form η  ξ = η 3 + Φ(ξ). Here, the function Φ(ξ)isdefined parametrically (x is the parameter) by the relations Φ = 1 f 3 U 3  f 0 – f 1 f 2 3f 3 + 2f 3 2 27f 2 3 + 1 3 d dx f 2 f 3  , ξ =  f 3 U 2 dx. 12.1.5-3. Reduction to an Abel equation of the second kind. Let y 0 = y 0 (x) be a particular solution of equation (12.1.5.1). Then the substitution y = y 0 + E(x) z(x) ,whereE(x)=exp   (3f 3 y 2 0 + 2f 2 y 0 + f 1 ) dx  , leads to an Abel equation of the second kind: zz  x =–(3f 3 y 0 + f 2 )Ez – f 3 E 2 . For equations of this type, see Subsection 12.1.6. 464 ORDINARY DIFFERENTIAL EQUATIONS 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 = f 2 (x)y 2 + f 1 (x)y + f 0 (x), g(x) 0.(12.1.6.1) The Abel equation (12.1.6.1) is not integrable for arbitrary f n (x)andg(x). Given below are some special cases where the Abel equation of the second kind is integrable by quadrature. 1 ◦ .Ifg(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–1 y + cx 2n–1 . See Paragraph 12.1.2-4 for k = n. The substitution w = yx –n leads to a separable equation: x(w + s)w  x =(a – n)w 2 +(b – ns)w + c. 4 ◦ . The Abel equation (y + a 2 x + c 2 )y  x = b 1 y + a 1 x + c 1 is a special case of the equation treated in Paragraph 12.1.2-3 (see Item 2 ◦ with f (w)=w and b 2 = 1). 5 ◦ . The unnormalized Abel equation [(a 1 x + a 2 x n )y + b 1 x + b 2 x n ]y  x = c 2 y 2 + c 1 y + c 0 can be reduced to the form (12.1.6.1) by dividing it by (a 1 x + a 2 x n ). Taking y to be the independent variable and x = x(y) to be the dependent one, we obtain the Bernoulli equation (c 2 y 2 + c 1 y + c 0 )x  y =(a 1 y + b 1 )x +(a 2 y + b 2 )x n . See Paragraph 12.1.2-6. 6 ◦ . The general solution of the Abel equation (y + g)y  x = f 2 y 2 + f 1 y + f 1 g – f 2 g 2 , f n = f n (x), g = g(x), is given by y =–g + CE + E  (f 1 + g  x – 2f 2 g)E –1 dx,whereE =exp   f 2 dx  . 7 ◦ .Iff 1 = 2f 2 g – g  x , the general solution of the Abel equation (12.1.6.1) has the form y =–g E  2  (f 0 + gg  x – f 2 g 2 )E –2 dx + C  1/2 ,whereE =exp   f 2 dx  . For other solvable Abel equations of the second kind, see the books by Kamke (1977) and Polyanin and Zaitsev (2003). . y) for some types of ordinary differential equations fdx+ gdy= 0,where f = f(x, y)andg = g(x, y). The subscripts x and y indicate the corresponding partial derivatives No. Conditions for f and. 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 = f 3 (x)y 3 +. 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)– f 2 3f 3 ,