486 ORDINARY DIFFERENTIAL EQUATIONS 2 ◦ . Suppose p(x)=s(x)=1 and the function q = q(x) has a continuous derivative. The following asymptotic relations hold for eigenvalues λ n and eigenfunctions y n (x)as n →∞: λ n = π(n – 1) x 2 – x 1 + 1 π(n – 1) Q(x 1 , x 2 )+O 1 n 2 , y n (x)=cos π(n – 1)(x – x 1 ) x 2 – x 1 + 1 π(n – 1) (x 1 – x)Q(x, x 2 ) +(x 2 – x)Q(x 1 , x) sin π(n – 1)(x – x 1 ) x 2 – x 1 + O 1 n 2 , where Q(u, v) is given by (12.2.5.8). 12.2.5-5. Problems with boundary conditions of the third kind. We consider the third boundary value problem for equation (12.2.5.1) subject to condi- tion (12.2.5.2) with α 1 = α 2 = 1. We assume that p(x)=s(x)=1 and the function q = q(x) has a continuous derivative. The following asymptotic formulas hold for eigenvalues λ n and eigenfunctions y n (x) as n →∞: λ n = π(n – 1) x 2 – x 1 + 1 π(n – 1) Q(x 1 , x 2 )–β 1 + β 2 + O 1 n 2 , y n (x)=cos π(n – 1)(x – x 1 ) x 2 – x 1 + 1 π(n – 1) (x 1 – x) Q(x, x 2 )+β 2 +(x 2 – x) Q(x 1 , x)–β 1 sin π(n – 1)(x – x 1 ) x 2 – x 1 + O 1 n 2 , where Q(u, v)isdefined by (12.2.5.8). 12.2.5-6. Problems with mixed boundary conditions. Let us note some special properties of the Sturm–Liouville problem that is the mixed boundary value problem for equation (12.2.5.1) with the boundary conditions y x = 0 at x = x 1 , y = 0 at x = x 2 . 1 ◦ .Ifq ≥ 0, the upper estimate (12.2.5.6) is valid for the least eigenvalue, with z = z(x) being any twice-differentiable function that satisfies the conditions z x (x 1 )=0 and z(x 2 )=0. The equality in (12.2.5.6) is attained if z = y 1 (x), where y 1 (x) is the eigenfunction corresponding to the eigenvalue λ 1 . 2 ◦ . Suppose p(x)=s(x)=1 and the function q = q(x) has a continuous derivative. The following asymptotic relations hold for eigenvalues λ n and eigenfunctions y n (x)as n →∞: λ n = π(2n – 1) 2(x 2 – x 1 ) + 2 π(2n – 1) Q(x 1 , x 2 )+O 1 n 2 , y n (x)=cos π(2n – 1)(x – x 1 ) 2(x 2 – x 1 ) + 2 π(2n – 1) (x 1 – x)Q(x, x 2 ) +(x 2 – x)Q(x 1 , x) sin π(2n – 1)(x – x 1 ) 2(x 2 – x 1 ) + O 1 n 2 , where Q(u, v)isdefined by (12.2.5.8). 12.2. SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS 487 12.2.6. Theorems on Estimates and Zeros of Solutions 12.2.6-1. Theorems on estimates of solutions. Let f n (x)andg n (x)(n = 1, 2) be continuous functions on the interval a ≤ x ≤ b and let the following inequalities hold: 0 ≤ f 1 (x) ≤ f 2 (x), 0 ≤ g 1 (x) ≤ g 2 (x). If y n = y n (x) are some solutions to the linear equations y n = f n (x)y n + g n (x)(n = 1, 2) and y 1 (a) ≤ y 2 (a)andy 1 (a) ≤ y 2 (a), then y 1 (x) ≤ y 2 (x)andy 1 (x) ≤ y 2 (x) on each interval a ≤ x ≤ a 1 ,wherey 2 (x)>0. 12.2.6-2. Sturm comparison theorem on zeros of solutions. Consider the equation [f(x)y ] + g(x)y = 0 (a ≤ x ≤ b), (12.2.6.1) where the function f(x) is positive and continuously differentiable, and the function g(x) is continuous. T HEOREM (COMPARISON,STURM). Let y n = y n (x) be nonzero solutions of the linear equations [f n (x)y n ] + g n (x)y n = 0 (n = 1, 2) and let the inequalities f 1 (x) ≥ f 2 (x)>0 and g 1 (x) ≤ g 2 (x) hold. Then the function y 2 has at least one zero lying between any two adjacent zeros, x 1 and x 2 , of the function y 1 (it is assumed that the identities f 1 ≡ f 2 and g 1 ≡ g 2 are not satisfied on any interval simultaneously). COROLLARY 1. If g(x) ≤ 0 or there exists a constant k 1 such that f(x) ≥ k 1 > 0, g(x)<k 1 π b – a 2 , then every nontrivial solution to equation (12.2.6.1) has no more than one zero on the interval [a, b] . COROLLARY 2. If there exists a constant k 2 such that 0 < f(x) ≤ k 2 , g(x)>k 2 πm b – a 2 , where m = 1, 2, , then every nontrivial solution to equation (12.2.6.1) has at least m zeros on the interval [a, b] . 488 ORDINARY DIFFERENTIAL EQUATIONS 12.2.6-3. Qualitative behavior of solutions as x →∞. Consider the equation y + f (x)y = 0,(12.2.6.2) where f (x) is a continuous function for x ≥ a. 1 ◦ .Forf (x) ≤ 0, every nonzero solution has no more than one zero, and hence y ≠ 0 for sufficiently large x. If f(x) ≤ 0 for all x and f(x) 0,theny ≡ 0 is the only solution bounded for all x. 2 ◦ . Suppose f(x) ≥ k 2 > 0. Then every nontrivial solution y(x) and its derivative y (x) have infinitely many zeros, with the distance between any adjacent zeros remaining finite. If f (x) → k 2 > 0 for x →∞and f ≥ 0, then the solutions of the equation for large x behave similarly to those of the equation y + k 2 y = 0. 3 ◦ .Letf (x) → –∞ for |x| →∞. Then every nonzero solution has only finitely many zeros, and |y /y| →∞as |x| →∞. There are two linearly independent solutions, y 1 and y 2 , such that y 1 → 0, y 1 → 0, y 2 →∞,andy 2 → –∞ as x → –∞, and there are two linearly independent solutions, ¯y 1 and ¯y 2 , such that ¯y 1 → 0, ¯y 1 → 0, ¯y 2 →∞,and¯y 2 →∞as x →∞. 4 ◦ . If the function f in equation (12.2.6.2) is continuous, monotonic, and positive, then the amplitude of each solution decreases (resp., increases) as f increases (resp., decreases). 12.3. Second-Order Nonlinear Differential Equations 12.3.1. Form of the General Solution. Cauchy Problem 12.3.1-1. Equations solved for the derivative. General solution. A second-order ordinary differential equation solved for the highest derivative has the form y xx = f (x, y, y x ). (12.3.1.1) The general solution of this equation depends on two arbitrary constants, C 1 and C 2 .In some cases, the general solution can be written in explicit form, y = ϕ(x, C 1 , C 2 ), but more often implicit or parametric forms of the general solution are encountered. 12.3.1-2. Cauchy problem. The existence and uniqueness theorem. Cauchy problem: Find a solution of equation (12.3.1.1) satisfying the initial conditions y(x 0 )=y 0 , y x (x 0 )=y 1 .(12.3.1.2) (At a point x = x 0 , the value of the unknown function, y 0 , and its derivative, y 1 ,are prescribed.) E XISTENCE AND UNIQUENESS THEOREM. Let f(x, y, z) be a continuous function in all its arguments in a neighborhood of a point (x 0 , y 0 , y 1 ) and let f have bounded par- tial derivatives f y and f z in this neighborhood, or the Lipschitz condition is satisfied: |f(x, y, z)–f(x, ¯y, ¯z)| ≤ A |y – ¯y| + |z – ¯z| , where A is some positive number. Then a solution of equation (12.3.1.1) satisfying the initial conditions (12.3.1.2) exists and is unique. 12.3. SECOND-ORDER NONLINEAR DIFFERENTIAL EQUATIONS 489 12.3.2. Equations Admitting Reduction of Order 12.3.2-1. Equations not containing y explicitly. In the general case, an equation that does not contain y implicitly has the form F (x, y x , y xx )=0.(12.3.2.1) Such equations remain unchanged under an arbitrary translation of the dependent variable: y → y + const. The substitution y x = z(x), y xx = z x (x) brings (12.3.2.1) to a first-order equation: F (x, z, z x )=0. 12.3.2-2. Equations not containing x explicitly (autonomous equations). In the general case, an equation that does not contain x implicitly has the form F (y, y x , y xx )=0.(12.3.2.2) Such equations remain unchanged under an arbitrary translation of the independent vari- able: x → x + const. Using the substitution y x = w(y), where y plays the role of the independent variable, and taking into account the relations y xx = w x = w y y x = w y w, one can reduce (12.3.2.2) to a first-order equation: F(y, w, ww y )=0. Example 1. Consider the autonomous equation y xx = f(y), which often arises in the theory of heat and mass transfer and combustion. The change of variable y x = w(y) leads to a separable first-order equation: ww y = f(y). Integrating yields w 2 = 2F (w)+C 1 ,where F (w)= f(w) dw. Solving for w and returning to the original variable, we obtain the separable equation y x = √ 2F (w)+C 1 . Its general solution is expressed as dy √ 2F (w)+C 1 = x + C 2 ,whereF (w)= f(w) dw. Remark. The equation y xx = f(y +ax 2 +bx+c) is reduced by the change of variable u = y +ax 2 +bx+c to an autonomous equation, u xx = f(u)+2a. 12.3.2-3. Equations of the form F (ax + by, y x , y xx )=0. Such equations are invariant under simultaneous translations of the independent and depen- dent variables in accordance with the rule x → x + bc, y → y – ac,where c is an arbitrary constant. For b = 0, see equation (12.3.2.1). For b ≠ 0, the substitution bw = ax + by leads to equation (12.3.2.2): F (bw, w x – a/b, w xx )=0. 12.3.2-4. Equations of the form F (x, xy x – y, y xx )=0. The substitution w(x)=xy x – y leads to a first-order equation: F (x, w, w x /x)=0. 490 ORDINARY DIFFERENTIAL EQUATIONS 12.3.2-5. Homogeneous equations. 1 ◦ .Theequations homogeneous in the independent variable remain unchanged under scaling of the independent variable, x → αx,whereα is an arbitrary nonzero number. In the general case, such equations can be written in the form F (y, xy x , x 2 y xx )=0.(12.3.2.3) The substitution z(y)=xy x leads to a first-order equation: F(y, z, zz y – z)=0. 2 ◦ .Theequations homogeneous in the dependent variable remain unchanged under scaling of the variable sought, y → αy,whereα is an arbitrary nonzero number. In the general case, such equations can be written in the form F (x, y x /y, y xx /y)=0.(12.3.2.4) The substitution z(x)=y x /y leads to a first-order equation: F (x, z, z x + z 2 )=0. 3 ◦ .Theequations homogeneous in both variables are invariant under simultaneous scaling (dilatation) of the independent and dependent variables, x →αx and y →αy,whereα is an arbitrary nonzero number. In the general case, such equations can be written in the form F (y/x, y x , xy xx )=0.(12.3.2.5) The transformation t =ln|x|, w = y/x leads to an autonomous equation (see Paragraph 12.3.2-2): F(w, w t + w, w tt + w t )=0. Example 2. The homogeneous equation xy xx – y x = f(y/x) is reduced by the transformation t =ln|x|, w = y/x to the autonomous form: w tt = f(w)+w. For solution of this equation, see Example 1 in Paragraph 12.3.2-2 (the notation of the right-hand side has to be changed there). 12.3.2-6. Generalized homogeneous equations. 1 ◦ .Thegeneralized homogeneous equations remain unchanged under simultaneous scaling of the independent and dependent variables in accordance with the rule x → αx and y → α k y,whereα is an arbitrary nonzero number and k is some number. Such equations can be written in the form F (x –k y, x 1–k y x , x 2–k y xx )=0.(12.3.2.6) The transformation t =lnx, w = x –k y leads to an autonomous equation (see Paragraph 12.3.2-2): F w, w t + kw, w tt +(2k – 1)w t + k(k – 1)w = 0. 2 ◦ . The most general form of representation of generalized homogeneous equations is as follows: F(x n y m , xy x /y, x 2 y xx /y)=0.(12.3.2.7) The transformation z = x n y m , u = xy x /y brings this equation to the fi rst-order equation F z, u, z(mu + n)u z – u + u 2 = 0. Remark. For m ≠ 0, equation (12.3.2.7) is equivalent to equation (12.3.2.6) in which k =–n/m.To the particular values n = 0 and m = 0 there correspond equations (12.3.2.3) and (12.3.2.4) homogeneous in the independent and dependent variables, respectively. For n =–m ≠ 0, we have an equation homogeneous in both variables, which is equivalent to equation (12.3.2.5). 12.3. SECOND-ORDER NONLINEAR DIFFERENTIAL EQUATIONS 491 12.3.2-7. Equations invariant under scaling–translation transformations. 1 ◦ . The equations of the form F (e λx y, e λx y x , e λx y xx )=0 (12.3.2.8) remain unchanged under simultaneous translation and scaling of variables, x → x + α and y → βy,whereβ = e –αλ and α is an arbitrary number. The substitution w = e λx y brings (12.3.2.8) to an autonomous equation: F (w, w x – λw, w xx – 2λw x + λ 2 w)=0 (see Paragraph 12.3.2-2). 2 ◦ . The equation F (e λx y n , y x /y, y xx /y)=0 (12.3.2.9) is invariant under the simultaneous translation and scaling of variables, x → x + α and y → βy,where β = e –αλ/n and α is an arbitrary number. The transformation z = e λx y n , w = y x /y brings (12.3.2.9) to a first-order equation: F z, w, z(nw + λ)w z + w 2 = 0. 3 ◦ . The equation F (x n e λy , xy x , x 2 y xx )=0 (12.3.2.10) is invariant under the simultaneous scaling and translation of variables, x → αx and y → y + β,where α =e –βλ/n and β is an arbitrary number. The transformation z = x n e λy , w = xy x brings (12.3.2.10) to a first-order equation: F z, w, z(λw + n)w z – w = 0. Some other second-order nonlinear equations are treated in Section T5.3. 12.3.2-8. Exact second-order equations. The second-order equation F (x, y, y x , y xx )=0 (12.3.2.11) is said tobe exact if itis thetotal differential of some function, F =ϕ x ,whereϕ=ϕ(x, y, y x ). If equation (12.3.2.11) is exact, then we have a first-order equation for y: ϕ(x, y, y x )=C,(12.3.2.12) where C is an arbitrary constant. If equation (12.3.2.11) is exact, then F (x, y, y x , y xx ) must have the form F (x, y, y x , y xx )=f(x, y, y x )y xx + g(x, y, y x ). (12.3.2.13) Here, f and g are expressed in terms of ϕ by the formulas f(x, y, y x )= ∂ϕ ∂y x , g(x, y, y x )= ∂ϕ ∂x + ∂ϕ ∂y y x .(12.3.2.14) By differentiating (12.3.2.14) with respect to x, y,and p = y x , we eliminate the variable ϕ from the two formulas in (12.3.2.14). As a result, we have the following test relations for f and g: f xx + 2pf xy + p 2 f yy = g xp + pg yp – g y , f xp + pf yp + 2f y = g pp . (12.3.2.15) Here, the subscripts denote the corresponding partial derivatives. 492 ORDINARY DIFFERENTIAL EQUATIONS If conditions (12.3.2.15) hold, then equation (12.3.2.11) with F of (12.3.2.13) is exact. In this case, we can integrate the first equation in (12.3.2.14) with respect to p = y x to determine ϕ = ϕ(x, y, y x ): ϕ = f(x, y, p) dp + ψ(x, y), (12.3.2.16) where ψ(x, y) is an arbitrary function of integration. This function is determined by substituting (12.3.2.16) into the second equation in (12.3.2.14). Example 3. The left-hand side of the equation yy xx +(y x ) 2 + 2axyy x + ay 2 = 0 (12.3.2.17) can be represented in the form (12.3.2.13), where f = y and g = p 2 + 2axyp + ay 2 . It is easy to verify that conditions (12.3.2.15) are satisfied. Hence, equation (12.3.2.17) is exact. Using (12.3.2.16), we obtain ϕ = yp + ψ(x, y). (12.3.2.18) Substituting this expression into the second equation in (12.3.2.14) and taking into account the relation g = p 2 + 2axyp + ay 2 ,wefind that 2axyp + ay 2 = ψ x + pψ y .Sinceψ = ψ(x, y), we have 2axy = ψ y and ay 2 = ψ x . Integrating yields ψ = axy 2 + const. Substituting this expression into (12.3.2.18) and taking into account relation (12.3.2.12), we find a first integral of equation (12.3.2.17): yp + axy 2 = C 1 ,wherep = y x . Setting w = y 2 , we arrive at the first-order linear equation w x + 2axw = 2C 1 , which is easy to integrate. Thus, we find the solution of the original equation in the form: y 2 = 2C 1 exp –ax 2 exp ax 2 dx + C 2 exp –ax 2 . 12.3.3. Methods of Regular Series Expansions with Respect to the Independent Variable 12.3.3-1. Method of expansion in powers of the independent variable. A solution of the Cauchy problem y xx = f (x, y, y x ), (12.3.3.1) y(x 0 )=y 0 , y x (x 0 )=y 1 (12.3.3.2) can be sought in the form of a Taylor series in powers of the difference (x–x 0 ), specifically: y(x)=y(x 0 )+y x (x 0 )(x – x 0 )+ y xx (x 0 ) 2! (x – x 0 ) 2 + y xxx (x 0 ) 3! (x – x 0 ) 3 + ···.(12.3.3.3) The first two coefficients y(x 0 )andy x (x 0 ) in solution (12.3.3.3) are defined by the initial conditions (12.3.3.2). The values of the subsequent derivatives of y at the point x = x 0 are determined from equation (12.3.3.1) and its derivative equations (obtained by successive differentiation of the equation) taking into account the initial conditions (12.3.3.2). In particular, setting x = x 0 in (12.3.3.1) and substituting (12.3.3.2), we obtain the value of the second derivative: y xx (x 0 )=f(x 0 , y 0 , y 1 ). (12.3.3.4) . finite. If f (x) → k 2 > 0 for x → and f ≥ 0, then the solutions of the equation for large x behave similarly to those of the equation y + k 2 y = 0. 3 ◦ .Letf (x) → –∞ for |x| →∞. Then every. reduced by the transformation t =ln|x|, w = y/x to the autonomous form: w tt = f(w)+w. For solution of this equation, see Example 1 in Paragraph 12.3.2-2 (the notation of the right-hand side has. transformations. 1 ◦ . The equations of the form F (e λx y, e λx y x , e λx y xx )=0 (12.3.2.8) remain unchanged under simultaneous translation and scaling of variables, x → x + α and y