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472 ORDINARY DIFFERENTIAL EQUATIONS 12.1.10-3. Runge–Kutta method of the fourth-order approximation. This is one of the widely used methods. The unknown values y k are successively found by the formulas y k+1 = y k + 1 6 (f 1 + 2f 2 + 2f 3 + f 4 )Δx, where f 1 = f (x k , y k ), f 2 = f (x k + 1 2 Δx, y k + 1 2 f 1 Δx), f 3 = f (x k + 1 2 Δx, y k + 1 2 f 2 Δx), f 4 = f (x k + Δx, y k + f 3 Δx). Remark 1. All methods described in Subsection 12.1.10 are special cases of the Runge–Kutta method (a detailed description of this method can be found in the monographs listed at the end of the current chapter). Remark 2. In practice, calculations are performed on the basis of any of the above recurrence formulas with two different steps Δx, 1 2 Δx and an arbitrarily chosen small Δx. Then one compares the results obtained at common points. If these results coincide within the given order of accuracy, one assumes that the chosen step Δx ensures the desired accuracy of calculations. Otherwise, the step is halved and the calculations are performed with the steps 1 2 Δx and 1 4 Δx, after which the results are compared again, etc. (Quite often, one compares the results of calculations with steps varying by ten or more times.) 12.2. Second-Order Linear Differential Equations 12.2.1. Formulas for the General Solution. Some Transformations 12.2.1-1. Homogeneous linear equations. Formulas for the general solution. 1 ◦ . Consider a second-order homogeneous linear equation in the general form f 2 (x)y xx + f 1 (x)y x + f 0 (x)y = 0.(12.2.1.1) The trivial solution, y = 0, is a particular solution of the homogeneous linear equation. Let y 1 (x), y 2 (x) be a fundamental system of solutions (nontrivial linearly independent particular solutions) of equation (12.2.1.1). Then the general solution is given by y = C 1 y 1 (x)+C 2 y 2 (x), (12.2.1.2) where C 1 and C 2 are arbitrary constants. 2 ◦ .Lety 1 = y 1 (x) be any nontrivial particular solution of equation (12.2.1.1). Then its general solution can be represented as y = y 1 C 1 + C 2 e –F y 2 1 dx ,whereF = f 1 f 2 dx.(12.2.1.3) 3 ◦ . Consider the equation y xx + f(x)y = 0, which is written in the canonical form; see Paragraph 12.2.1-3 for the reduction of equations to this form. Let y 1 (x) be any nontrivial partial solution of this equation. The general solution can be constructed by formula (12.2.1.3) with F = 0 or formula (12.2.1.2) in which y 2 (x)=y 1 [f(x)–1][y 2 1 –(y 1 ) 2 ] [y 2 1 +(y 1 ) 2 ] 2 dx + y 1 y 2 1 +(y 1 ) 2 . Here, y 1 = y 1 (x) and the prime denotes differentiation with respect to x. The last formula is suitable where y 1 vanishes at some points. 12.2. SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS 473 4 ◦ . The second-order constant coefficient linear equation y xx + ay x + by = 0 has the following fundamental system of solutions: y 1 (x)=exp – 1 2 ax sinh 1 2 x √ a 2 – 4b , y 2 (x)=exp – 1 2 ax cosh 1 2 x √ a 2 – 4b if a 2 > 4b; y 1 (x)=exp – 1 2 ax sin 1 2 x √ 4b – a 2 , y 2 (x)=exp – 1 2 ax cos 1 2 x √ 4b – a 2 if a 2 < 4b; y 1 (x)=exp – 1 2 ax , y 2 (x)=x exp – 1 2 ax if a 2 = 4b. 5 ◦ .TheEuler equation x 2 y xx + axy x + by = 0 is reduced by the change of variable x = ke t (k ≠ 0) to the second-order constant coefficient linear equation y tt +(a – 1)y t + by = 0, which is treated in Item 4 ◦ . Solutions to some other second-order linear equations can be found in Section T5.2. 12.2.1-2. Wronskian determinant and Liouville’s formula. The Wronskian determinant (or Wronskian)isdefined by W (x)= y 1 (x) y 2 (x) y 1 (x) y 2 (x) ≡ y 1 (y 2 ) x – y 2 (y 1 ) x , where y 1 (x), y 2 (x) is a fundamental system of solutions of equation (12.2.1.1). Liouville’s formula: W (x)=W (x 0 )exp – x x 0 f 1 (t) f 2 (t) dt . 12.2.1-3. Reduction to the canonical form. 1 ◦ . The substitution y = u(x)exp – 1 2 f 1 f 2 dx (12.2.1.4) brings equation (12.2.1.1) to the canonical (or normal) form u xx + f(x)u = 0,wheref = f 0 f 2 – 1 4 f 1 f 2 2 – 1 2 f 1 f 2 x .(12.2.1.5) 2 ◦ . The substitution (12.2.1.4) is a special case of the more general transformation (ϕ is an arbitrary function) x = ϕ(ξ), y = u(ξ) |ϕ ξ (ξ)| exp – 1 2 f 1 (ϕ) f 2 (ϕ) dϕ , which also brings the original equation to the canonical form. 474 ORDINARY DIFFERENTIAL EQUATIONS 12.2.1-4. Reduction to the Riccati equation. The substitution u = y x /y brings the second-order homogeneous linear equation (12.2.1.1) to the Riccati equation: f 2 (x)u x + f 2 (x)u 2 + f 1 (x)u + f 0 (x)=0, which is discussed in Subsection 12.1.4. 12.2.1-5. Nonhomogeneous linear equations. The existence theorem. A second-order nonhomogeneous linear equation has the form f 2 (x)y xx + f 1 (x)y x + f 0 (x)y = g(x). (12.2.1.6) T HEOREM (EXISTENCE AND UNIQUENESS). On an open interval a < x < b, let the functions f 2 , f 1 , f 0 ,and g be continuous and f 2 ≠ 0 .Alsolet y(x 0 )=A, y x (x 0 )=B be arbitrary initial conditions, where x 0 is any point such that a < x 0 < b, and A and B are arbitrary prescribed numbers. Then a solution of equation (12.2.1.6) exists and is unique. This solutions is defined for all x (a, b) . 12.2.1-6. Nonhomogeneous linear equations. Formulas for the general solution. 1 ◦ . The general solution of the nonhomogeneous linear equation (12.2.1.6) is the sum of the general solution of the corresponding homogeneous equation (12.2.1.1) and any particular solution of the nonhomogeneous equation (12.2.1.6). 2 ◦ .Lety 1 = y 1 (x), y 2 = y 2 (x) be a fundamental system of solutions of the corresponding homogeneous equation, with g ≡ 0. Then the general solution of equation (12.2.1.6) can be represented as y = C 1 y 1 + C 2 y 2 + y 2 y 1 g f 2 dx W – y 1 y 2 g f 2 dx W ,(12.2.1.7) where W = y 1 (y 2 ) x – y 2 (y 1 ) x is the Wronskian determinant. 3 ◦ . Given a nontrivial particular solution y 1 = y 1 (x) of the homogeneous equation (with g ≡ 0), a second particular solution y 2 = y 2 (x) can be calculated from the formula y 2 = y 1 e –F y 2 1 dx,whereF = f 1 f 2 dx, W = e –F .(12.2.1.8) Then the general solution of equation (12.2.1.6) can be constructed by (12.2.1.7). 4 ◦ .Let¯y 1 and ¯y 2 be respective solutions of the nonhomogeneous differential equations L[¯y 1 ]=g 1 (x)andL[¯y 2 ]=g 2 (x), which have the same left-hand side but different right- hand sides, where L [y] is the left-hand side of equation (12.2.1.6). Then the function ¯y = ¯y 1 + ¯y 2 is a solution of the equation L [¯y]=g 1 (x)+g 2 (x). 12.2. SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS 475 12.2.1-7. Reduction to a constant coefficient equation (a special case). Let f 2 = 1, f 0 ≠ 0, and the condition 1 |f 0 | d dx |f 0 | + f 1 √ |f 0 | = a = const be satisfied. Then the substitution ξ = |f 0 | dx leads to a constant coefficient linear equation, y ξξ + ay ξ + y sign f 0 = 0. 12.2.1-8. Kummer–Liouville transformation. The transformation x = α(t), y = β(t)z + γ(t), (12.2.1.9) where α(t), β(t), and γ(t) are arbitrary sufficiently smooth functions (β 0), takes any linear differential equation for y(x) to a linear equation for z = z(t). In the special case γ ≡ 0, a homogeneous equation is transformed to a homogeneous one. Special cases of transformation (12.2.1.9) are widely used to simplify second- and higher-order linear differential equations. 12.2.2. Representation of Solutions as a Series in the Independent Variable 12.2.2-1. Equation coefficients are representable in the ordinary power series form. Let us consider a homogeneous linear differential equation of the general form y xx + f(x)y x + g(x)y = 0.(12.2.2.1) Assume that the functions f (x)andg(x) are representable, in the vicinity of a point x = x 0 , in the power series form, f(x)= ∞ n=0 A n (x – x 0 ) n , g(x)= ∞ n=0 B n (x – x 0 ) n ,(12.2.2.2) on the interval |x – x 0 | < R,whereR stands for the minimum radius of convergence of the two series in (12.2.2.2). In this case, the point x = x 0 is referred to as an ordinary point, and equation (12.2.2.1) possesses two linearly independent solutions of the form y 1 (x)= ∞ n=0 a n (x – x 0 ) n , y 2 (x)= ∞ n=0 b n (x – x 0 ) n .(12.2.2.3) The coefficients a n and b n are determined by substituting the series (12.2.2.2) into equa- tion (12.2.2.1) followed by extracting the coefficients of like powers of (x – x 0 ).* * Prior to that, the terms containing the same powers (x – x 0 ) k , k = 0, 1, , should be collected. 476 ORDINARY DIFFERENTIAL EQUATIONS 12.2.2-2. Equation coefficients have poles at some point. Assume that the functions f(x)andg(x) are representable, in the vicinity of a point x = x 0 , in the form f(x)= ∞ n=–1 A n (x – x 0 ) n , g(x)= ∞ n=–2 B n (x – x 0 ) n ,(12.2.2.4) on the interval |x–x 0 | < R. In this case, the point x = x 0 is referred to as a regular singular point.Letλ 1 and λ 2 be roots of the quadratic equation λ 2 1 +(A –1 – 1)λ + B –2 = 0. There are three cases, depending on the values of the exponents of the singularity. 1. If λ 1 ≠ λ 2 and λ 1 – λ 2 is not an integer, equation (12.2.2.1) has two linearly independent solutions of the form y 1 (x)=|x – x 0 | λ 1 1 + ∞ n=1 a n (x – x 0 ) n , y 2 (x)=|x – x 0 | λ 2 1 + ∞ n=1 b n (x – x 0 ) n . (12.2.2.5) 2. If λ 1 = λ 2 = λ, equation (12.2.2.1) possesses two linearly independent solutions: y 1 (x)=|x – x 0 | λ 1 + ∞ n=1 a n (x – x 0 ) n , y 2 (x)=y 1 (x)ln|x – x 0 | + |x – x 0 | λ ∞ n=0 b n (x – x 0 ) n . 3. If λ 1 = λ 2 + N,whereN is a positive integer, equation (12.2.2.1) has two linearly independent solutions of the form y 1 (x)=|x – x 0 | λ 1 1 + ∞ n=1 a n (x – x 0 ) n , y 2 (x)=ky 1 (x)ln|x – x 0 | + |x – x 0 | λ 2 ∞ n=0 b n (x – x 0 ) n , where k is a constant to be determined (it may be equal to zero). To construct the solution in each of the three cases, the following procedure should be performed: substitute the above expressions of y 1 and y 2 into the original equa- tion (12.2.2.1) and equate the coefficients of (x – x 0 ) n and (x – x 0 ) n ln |x–x 0 | for different values of n to obtain recurrence relations for the unknown coefficients. From these recur- rence relations the solution sought can be found. 12.2. SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS 477 12.2.3. Asymptotic Solutions This subsection presents asymptotic solutions, as ε → 0 (ε > 0), of some second-order linear ordinary differential equations containing arbitrary functions (sufficiently smooth), with the independent variable being real. 12.2.3-1. Equations not containing y x . Leading asymptotic terms. 1 ◦ . Consider the equation ε 2 y xx – f (x)y = 0 (12.2.3.1) on a closed interval a ≤ x ≤ b. Case 1. With the condition f ≠ 0, the leading terms of the asymptotic expansions of the fundamental system of solutions, as ε → 0, are given by the formulas y 1 = f –1/4 exp – 1 ε fdx , y 2 = f –1/4 exp 1 ε fdx if f > 0, y 1 =(–f) –1/4 cos 1 ε –fdx , y 2 =(–f) –1/4 sin 1 ε –fdx if f < 0. Case 2. Discuss the asymptotic solution of equation (12.2.3.1) in the vicinity of the point x = x 0 , where function f (x)vanishes,f(x 0 )=0 (such a point is referred to as a transition point). We assume that the function f can be presented in the form f(x)=(x 0 – x)ψ(x), where ψ(x)>0. In this case, the fundamental solutions, as ε →0, are described by three different formulas: y 1 = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 |f(x)| 1/4 sin 1 ε x x 0 |f(x)| dx + π 4 if x – x 0 ≥ δ, √ π [εψ(x 0 )] 1/6 Ai(z)if|x – x 0 | ≤ δ, 1 2[f(x)] 1/4 exp – 1 ε x 0 x f(x) dx if x 0 – x ≥ δ, y 2 = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 |f(x)| 1/4 cos 1 ε x x 0 |f(x)| dx + π 4 if x – x 0 ≥ δ, √ π [εψ(x 0 )] 1/6 Bi(z)if|x – x 0 | ≤ δ, 1 [f(x)] 1/4 exp 1 ε x 0 x f(x) dx if x 0 – x ≥ δ, where Ai(z)andBi(z) are the Airy functions of the first and second kind, respectively (see Section 18.8), z = ε –2/3 [ψ(x 0 )] 1/3 (x 0 – x), and δ = O(ε 2/3 ). 478 ORDINARY DIFFERENTIAL EQUATIONS 12.2.3-2. Equations not containing y x . Two-term asymptotic expansions. The two-term asymptotic expansions of the solution of equation (12.2.3.1) with f > 0,as ε → 0, on a closed interval a ≤ x ≤ b,hastheform y 1 = f –1/4 exp – 1 ε x x 0 fdx 1 – ε x x 0 1 8 f xx f 3/2 – 5 32 (f x ) 2 f 5/2 dx + O(ε 2 ) , y 2 = f –1/4 exp 1 ε x x 0 fdx 1 + ε x x 0 1 8 f xx f 3/2 – 5 32 (f x ) 2 f 5/2 dx + O(ε 2 ) , (12.2.3.2) where x 0 is an arbitrary number satisfying the inequality a ≤ x 0 ≤ b. The asymptotic expansions of the fundamental system of solutions of equation (12.2.3.1) with f < 0 are derived by separating the real and imaginary parts in either formula (12.2.3.2). 12.2.3-3. Equations of special form not containing y x . Consider the equation ε 2 y xx – x m–2 f(x)y = 0 (12.2.3.3) on a closed interval a ≤ x ≤ b,wherea<0 and b >0, under the conditions that m is a positive integer and f (x) ≠ 0. In this case, the leading term of the asymptotic solution, as ε → 0, in the vicinity of the point x = 0 is expressed in terms of a simpler model equation, which results from substituting the function f(x) in equation (12.2.3.3) by the constant f(0)(the solution of the model equation is expressed in terms of the Bessel functions of order 1/m). We specify below formulas by which the leading terms of the asymptotic expansions of the fundamental system of solutions of equation (12.2.3.3) with a < x < 0 and 0 < x < b are related (excluding a small vicinity of the point x = 0). Three different cases can be extracted. 1 ◦ .Letm be an even integer and f (x)>0. Then, y 1 = ⎧ ⎪ ⎨ ⎪ ⎩ [f(x)] –1/4 exp 1 ε x 0 f(x) dx if x < 0, k –1 [f(x)] –1/4 exp 1 ε x 0 f(x) dx if x > 0, y 2 = ⎧ ⎪ ⎨ ⎪ ⎩ [f(x)] –1/4 exp – 1 ε x 0 f(x) dx if x < 0, k[f(x)] –1/4 exp – 1 ε x 0 f(x) dx if x > 0, where f = f(x), k =sin π m . 2 ◦ .Letm be an even integer and f (x)<0. Then, y 1 = ⎧ ⎪ ⎨ ⎪ ⎩ |f(x)| –1/4 cos – 1 ε x 0 |f(x)| dx + π 4 if x < 0, k –1 |f(x)| –1/4 cos 1 ε x 0 |f(x)| dx – π 4 if x > 0, y 2 = ⎧ ⎪ ⎨ ⎪ ⎩ |f(x)| –1/4 cos – 1 ε x 0 |f(x)| dx – π 4 if x < 0, k|f(x)| –1/4 cos 1 ε x 0 |f(x)| dx + π 4 if x > 0, . canonical form; see Paragraph 12.2.1-3 for the reduction of equations to this form. Let y 1 (x) be any nontrivial partial solution of this equation. The general solution can be constructed by formula. expansions of the fundamental system of solutions of equation (12.2.3.1) with f < 0 are derived by separating the real and imaginary parts in either formula (12.2.3.2). 12.2.3-3. Equations of special. each of the three cases, the following procedure should be performed: substitute the above expressions of y 1 and y 2 into the original equa- tion (12.2.2.1) and equate the coefficients of (x
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