12.2. SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS 479 where f = f(x), k =tan π 2m . 3 ◦ .Letm be an odd integer. Then, y 1 = ⎧ ⎪ ⎨ ⎪ ⎩ |f(x)| –1/4 cos – 1 ε x 0 |f(x)| dx + π 4 if x < 0, 1 2 k –1 [f(x)] –1/4 exp 1 ε x 0 f(x) dx 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 exp – 1 ε x 0 f(x) dx if x > 0, where f = f(x), k =sin π 2m . 12.2.3-4. Equations not containing y x . Equation coefficients are dependent on ε. Consider an equation of the form ε 2 y xx – f(x, ε)y = 0 (12.2.3.4) on a closed interval a ≤ x ≤ b under the condition that f ≠ 0. Assume that the following asymptotic relation holds: f(x, ε)= ∞ k=0 f k (x)ε k , ε → 0. Then the leading terms of the asymptotic expansions of the fundamental system of solutions of equation (12.2.3.4) are given by the formulas y 1 = f –1/4 0 (x)exp – 1 ε f 0 (x) dx + 1 2 f 1 (x) √ f 0 (x) dx 1 + O(ε) , y 2 = f –1/4 0 (x)exp 1 ε f 0 (x) dx + 1 2 f 1 (x) √ f 0 (x) dx 1 + O(ε) . 12.2.3-5. Equations containing y x . 1 ◦ . Consider an equation of the form εy xx + g(x)y x + f(x)y = 0 on a closed interval 0 ≤ x ≤ 1. With g(x)>0, the asymptotic solution of this equation, satisfying the boundary conditions y(0)=C 1 and y(1)=C 2 , can be represented in the form y =(C 1 – kC 2 )exp –ε –1 g(0)x + C 2 exp 1 x f(x) g(x) dx + O(ε), where k =exp 1 0 f(x) g(x) dx . 480 ORDINARY DIFFERENTIAL EQUATIONS 2 ◦ . Now let us take a look at an equation of the form ε 2 y xx + εg(x)y x + f(x)y = 0 (12.2.3.5) on a closed interval a ≤ x ≤ b. Assume D(x) ≡ [g(x)] 2 – 4f(x) ≠ 0. Then the leading terms of the asymptotic expansions of the fundamental system of solutions of equation (12.2.3.5), as ε → 0, are expressed by y 1 = |D(x)| –1/4 exp – 1 2ε D(x) dx – 1 2 g x (x) √ D(x) dx 1 + O(ε) , y 2 = |D(x)| –1/4 exp 1 2ε D(x) dx – 1 2 g x (x) √ D(x) dx 1 + O(ε) . 12.2.3-6. Equations of the general form. The more general equation ε 2 y xx + εg(x, ε)y x + f(x, ε)y = 0 is reducible, with the aid of the substitution y = w exp – 1 2ε gdx , to an equation of the form (12.2.3.4), ε 2 w xx +(f – 1 4 g 2 – 1 2 εg x )w = 0, to which the asymptotic formulas given above in Paragraph 12.2.3-4 are applicable. 12.2.4. Boundary Value Problems 12.2.4-1. First, second, third, and mixed boundary value problems (x 1 ≤ x ≤ x 2 ). We consider the second-order nonhomogeneous linear differential equation y xx + f(x)y x + g(x)y = h(x). (12.2.4.1) 1 ◦ . The first boundary value problem: Find a solution of equation (12.2.4.1) satisfying the boundary conditions y = a 1 at x = x 1 , y = a 2 at x = x 2 .(12.2.4.2) (The values of the unknown are prescribed at two distinct points x 1 and x 2 .) 2 ◦ . The second boundary value problem: Find a solution of equation (12.2.4.1) satisfying the boundary conditions y x = a 1 at x = x 1 , y x = a 2 at x = x 2 .(12.2.4.3) (The values of the derivative of the unknown are prescribed at two distinct points x 1 and x 2 .) 12.2. SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS 481 3 ◦ . The third boundary value problem: Find a solution of equation (12.2.4.1) satisfying the boundary conditions y x – k 1 y = a 1 at x = x 1 , y x + k 2 y = a 2 at x = x 2 . (12.2.4.4) 4 ◦ . The third boundary value problem: Find a solution of equation (12.2.4.1) satisfying the boundary conditions y = a 1 at x = x 1 , y x = a 2 at x = x 2 .(12.2.4.5) (The unknown itself is prescribed at one point, and its derivative at another point.) Conditions (12.2.4.2), (12.2.4.3), (12.2.4.4), and (12.2.4.5) are called homogeneous if a 1 = a 2 = 0. 12.2.4-2. Simplification of boundary conditions. The self-adjoint form of equations. 1 ◦ . Nonhomogeneous boundary conditions can be reduced to homogeneous ones by the change of variable z =A 2 x 2 +A 1 x+A 0 +y (the constants A 2 , A 1 ,and A 0 are selected using the method of undetermined coefficients). In particular, the nonhomogeneous boundary conditions of the first kind (12.2.4.2) can be reduced to homogeneous boundary conditions by the linear change of variable z = y – a 2 – a 1 x 2 – x 1 (x – x 1 )–a 1 . 2 ◦ . On multiplying by p(x)=exp f(x) dx , one reduces equation (12.2.4.1) to the self-adjoint form: [p(x)y x ] x + q(x)y = r(x). (12.2.4.6) Without loss of generality, we can further consider equation (12.2.4.6) instead of (12.2.4.1). We assume that the functions p, p x , q,andr are continuous on the inter- val x 1 ≤ x ≤ x 2 ,andp is positive. 12.2.4-3. Green’s function. Linear problems for nonhomogeneous equations. The Green’s function of the first boundary value problem for equation (12.2.4.6) with homogeneous boundary conditions (12.2.4.2) is a function of two variables G(x, s)that satisfies the following conditions: 1 ◦ . G(x, s) is continuous in x for fixed s, with x 1 ≤ x ≤ x 2 and x 1 ≤ s ≤ x 2 . 2 ◦ . G(x, s) is a solution of the homogeneous equation (12.2.4.6), with r = 0,forall x 1 < x < x 2 exclusive of the point x = s. 3 ◦ . G(x, s) satisfies the homogeneous boundary conditions G(x 1 , s)=G(x 2 , s)=0. 4 ◦ . The derivative G x (x, s)hasajumpof1/p(s) at the point x = s,thatis, G x (x, s) x→s, x>s – G x (x, s) x→s, x<s = 1 p(s) . For the second, third, and mixed boundary value problems, the Green’s function is de- fined likewise except that in 3 ◦ the homogeneous boundary conditions (12.2.4.3), (12.2.4.4), and (12.2.4.5), with a 1 = a 2 = 0, are adopted, respectively. 482 ORDINARY DIFFERENTIAL EQUATIONS The solution of the nonhomogeneous equation (12.2.4.6) subject to appropriate homo- geneous boundary conditions is expressed in terms of the Green’s function as follows:* y(x)= x 2 x 1 G(x, s)r(s) ds. 12.2.4-4. Representation of the Green’s function in terms of particular solutions. We consider the first boundary value problem. Let y 1 = y 1 (x)andy 2 = y 2 (x) be linearly independent particular solutions of the homogeneous equation (12.2.4.6), with r = 0,that satisfy the conditions y 1 (x 1 )=0, y 2 (x 2 )=0. (Each of the solutions satisfies one of the homogeneous boundary conditions.) The Green’s function is expressed in terms of solutions of the homogeneous equation as follows: G(x, s)= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ y 1 (x)y 2 (s) p(s)W (s) for x 1 ≤ x ≤ s, y 1 (s)y 2 (x) p(s)W (s) for s ≤ x ≤ x 2 , (12.2.4.7) where W (x)=y 1 (x)y 2 (x)–y 1 (x)y 2 (x) is the Wronskian determinant. Remark. Formula (12.2.4.7) can also be used to construct the Green’s functions for the second, third, and mixed boundary value problems. To this end, one should find two linearly independent solutions, y 1 = y 1 (x) and y 2 = y 2 (x), of the homogeneous equation; the former satisfies the corresponding homogeneous boundary condition at x = x 1 and the latter satisfies the one at x = x 2 . 12.2.5. Eigenvalue Problems 12.2.5-1. Sturm–Liouville problem. Consider the second-order homogeneous linear differential equation [p(x)y x ] x +[λs(x)–q(x)]y = 0 (12.2.5.1) subject to linear boundary conditions of the general form α 1 y x + β 1 y = 0 at x = x 1 , α 2 y x + β 2 y = 0 at x = x 2 . (12.2.5.2) It is assumed that the functions p, p x , s,and q are continuous, and p and s are positive on an interval x 1 ≤ x ≤ x 2 . It is also assumed that |α 1 | + |β 1 | > 0 and |α 2 | + |β 2 | > 0. The Sturm–Liouville problem: Find the values λ n of the parameter λ at which problem (12.2.5.1), (12.2.5.2) has a nontrivial solution. Such λ n are called eigenvalues and the cor- responding solutions y n = y n (x) are called eigenfunctions of the Sturm–Liouville problem (12.2.5.1), (12.2.5.2). * The homogeneous boundary value problem—with r(x)=0 and a 1 = a 2 = 0—is assumed to have only the trivial solution. 12.2. SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS 483 12.2.5-2. General properties of the Sturm–Liouville problem (12.2.5.1), (12.2.5.2). 1 ◦ .Thereareinfinitely (countably) many eigenvalues. All eigenvalues can be ordered so that λ 1 < λ 2 < λ 3 < ···. Moreover, λ n →∞as n →∞; hence, there can only be a finite number of negative eigenvalues. Each eigenvalue has multiplicity 1. 2 ◦ . The eigenfunctions are defined up to a constant factor. Each eigenfunction y n (x)has precisely n – 1 zeros on the open interval (x 1 , x 2 ). 3 ◦ . Any two eigenfunctions y n (x)andy m (x), n ≠ m, are orthogonal with weight s(x) on the interval x 1 ≤ x ≤ x 2 : x 2 x 1 s(x)y n (x)y m (x) dx = 0 if n ≠ m. 4 ◦ . An arbitrary function F (x) that has a continuous derivative and satisfies the boundary conditions of the Sturm–Liouville problem can be decomposed into an absolutely and uniformly convergent series in the eigenfunctions F (x)= ∞ n=1 F n y n (x), where the Fourier coefficients F n of F (x) are calculated by F n = 1 y n 2 x 2 x 1 s(x)F (x)y n (x) dx, y n 2 = x 2 x 1 s(x)y 2 n (x) dx. 5 ◦ . If the conditions q(x) ≥ 0, α 1 β 1 ≤ 0, α 2 β 2 ≥ 0 (12.2.5.3) hold true, there are no negative eigenvalues. If q ≡ 0 and β 1 = β 2 = 0, the least eigenvalue is λ 1 = 0, to which there corresponds an eigenfunction y 1 = const. In the other cases where conditions (12.2.5.3) are satisfied, all eigenvalues are positive. 6 ◦ . The following asymptotic formula is valid for eigenvalues as n →∞: λ n = π 2 n 2 Δ 2 + O(1), Δ = x 2 x 1 s(x) p(x) dx.(12.2.5.4) Paragraphs 12.2.5-3 through 12.2.5-6 will describe special properties of the Sturm– Liouville problem that depend on the specific form of the boundary conditions. Remark 1. Equation (12.2.5.1) can be reduced to the case where p(x) ≡ 1 and s(x) ≡ 1 by the change of variables ζ = s(x) p(x) dx, u(ζ)= p(x)s(x) 1/4 y(x). In this case, the boundary conditions are transformed to boundary conditions of similar form. Remark 2. The second-order linear equation ϕ 2 (x)y xx + ϕ 1 (x)y x +[λ + ϕ 0 (x)]y = 0 can be represented in the form of equation (12.2.5.1) where p(x), s(x), and q(x)aregivenby p(x)=exp ϕ 1 (x) ϕ 2 (x) dx , s(x)= 1 ϕ 2 (x) exp ϕ 1 (x) ϕ 2 (x) dx , q(x)=– ϕ 0 (x) ϕ 2 (x) exp ϕ 1 (x) ϕ 2 (x) dx . 484 ORDINARY DIFFERENTIAL EQUATIONS TABLE 12.2 Example estimates of the first eigenvalue λ 1 in Sturm–Liouville problems with boundary conditions of the first kind y(0)=y(1)=0 obtained using the Rayleigh–Ritz principle [the right-hand side of relation (12.2.5.6)] Equation Test function λ 1 , approximate λ 1 , exact y xx + λ(1 + x 2 ) –2 y = 0 z =sinπx 15.337 15.0 y xx + λ(4 – x 2 ) –2 y = 0 z =sinπx 135.317 134.837 [(1 + x) –1 y x ] x + λy = 0 z =sinπx 7.003 6.772 √ 1 + xy x x + λy = 0 z =sinπx 11.9956 11.8985 y xx + λ(1 +sinπx)y = 0 z =sinπx z = x(1 – x) 0.54105 π 2 0.55204 π 2 0.54032 π 2 0.54032 π 2 12.2.5-3. Problems with boundary conditions of the first kind. Let us note some special properties of the Sturm–Liouville problem that is the first boundary value problem for equation (12.2.5.1) with the boundary conditions y = 0 at x = x 1 , y = 0 at x = x 2 .(12.2.5.5) 1 ◦ .Forn →∞, the asymptotic relation (12.2.5.4) can be used to estimate the eigenval- ues λ n . In this case, the asymptotic formula y n (x) y n = 4 Δ 2 p(x)s(x) 1/4 sin πn Δ x x 1 s(x) p(x) dx + O 1 n , Δ = x 2 x 1 s(x) p(x) dx holds true for the eigenfunctions y n (x). 2 ◦ .Ifq ≥ 0, the following upper estimate holds for the least eigenvalue (Rayleigh–Ritz principle): λ 1 ≤ x 2 x 1 p(x)(z x ) 2 + q(x)z 2 dx x 2 x 1 s(x)z 2 dx ,(12.2.5.6) where z = z(x) is any twice differentiable function that satisfies the conditions z(x 1 )= z(x 2 )=0. The equalityin (12.2.5.6) isattained if z =y 1 (x), where y 1 (x) is the eigenfunction corresponding to the eigenvalue λ 1 . One can take z =(x–x 1 )(x 2 –x)orz =sin π(x – x 1 ) x 2 – x 1 in (12.2.5.6) to obtain specific estimates. It is significant to note that the left-hand side of (12.2.5.6) usually gives a fairly precise estimate of the first eigenvalue (see Table 12.2). 3 ◦ . The extension of the interval [x 1 , x 2 ] leads to decreasing in eigenvalues. 4 ◦ . Let the inequalities 0 < p min ≤ p(x) ≤ p max , 0 < s min ≤ s(x) ≤ s max , 0 < q min ≤ q(x) ≤ q max be satisfied. Then the following bilateral estimates hold: p min s max π 2 n 2 (x 2 – x 1 ) 2 + q min s max ≤ λ n ≤ p max s min π 2 n 2 (x 2 – x 1 ) 2 + q max s min . 12.2. SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS 485 5 ◦ . In engineering calculations for eigenvalues, the approximate formula λ n = π 2 n 2 Δ 2 + 1 x 2 – x 1 x 2 x 1 q(x) s(x) dx, Δ = x 2 x 1 s(x) p(x) dx (12.2.5.7) may be quite useful. This formula provides an exact result if p(x)s(x) = const and q(x)/s(x) = const (in particular, for constant equation coefficients, p = p 0 , q = q 0 ,and s = s 0 ) and gives a correct asymptotic behavior of (12.2.5.4) for any p(x), q(x), and s(x). In addition, relation (12.2.5.7) gives two correct leading asymptotic terms as n →∞if p(x) = const and s(x)=const [andalsoif p(x)s(x) = const]. 6 ◦ . 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 = πn x 2 – x 1 + 1 πn Q(x 1 , x 2 )+O 1 n 2 , y n (x)=sin πn(x – x 1 ) x 2 – x 1 – 1 πn (x 1 – x)Q(x, x 2 )+(x 2 – x)Q(x 1 , x) cos πn(x – x 1 ) x 2 – x 1 + O 1 n 2 , where Q(u, v)= 1 2 v u q(x) dx.(12.2.5.8) 7 ◦ . Let us consider the eigenvalue problem for the equation with a small parameter y xx +[λ + εq(x)]y = 0 (ε → 0) subject to the boundary conditions (12.2.5.5) with x 1 = 0 and x 2 = 1. We assume that q(x)=q(–x). This problem has the following eigenvalues and eigenfunctions: λ n = π 2 n 2 – εA nn + ε 2 π 2 k≠n A 2 nk n 2 – k 2 + O(ε 3 ), A nk = 2 1 0 q(x)sin(πnx)sin(πkx) dx; y n (x)= √ 2 sin(πnx)–ε √ 2 π 2 k≠n A nk n 2 – k 2 sin(πkx)+O(ε 2 ). Here, the summation is carried out over k from 1 to ∞. The next term in the expansion of y n can be found in Nayfeh (1973). 12.2.5-4. Problems with boundary conditions of the second kind. Let us note some special properties of the Sturm–Liouville problem that is the second boundary value problem for equation (12.2.5.1) with the boundary conditions y x = 0 at x = x 1 , y x = 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 )=z x (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 . . const and q(x)/s(x) = const (in particular, for constant equation coefficients, p = p 0 , q = q 0 ,and s = s 0 ) and gives a correct asymptotic behavior of (12.2.5.4) for any p(x), q(x), and s(x). In. Equations of the general form. The more general equation ε 2 y xx + εg(x, ε)y x + f(x, ε)y = 0 is reducible, with the aid of the substitution y = w exp – 1 2ε gdx , to an equation of the form. the change of variable z =A 2 x 2 +A 1 x+A 0 +y (the constants A 2 , A 1 ,and A 0 are selected using the method of undetermined coefficients). In particular, the nonhomogeneous boundary conditions of