Handbook of mathematics for engineers and scienteists part 80 ppt

12.4. LINEAR EQUATIONS OF ARBITRARY ORDER 521 Example 1. Consider a special case of equation (12.4.2.7): xy  xx + y  x + axy = 0.(12.4.2.8) Denote y(0)=y 0 and y  x (0)=y 1 . Let us apply the Laplace transform to this equation using formulas (12.4.2.6). On rearrangement, we obtain a linear first-order equation for y(p): –(p 2 y – y 0 p – y 1 )  p +(py – y 0 )–ay  p = 0 =⇒ (p 2 + a)y  p + py = 0. Its general solution is expressed as y = C  p 2 + a ,(12.4.2.9) where C is an arbitrary constant. Applying the inverse Laplace transform to (12.4.2.9) and taking into account formulas 19 and 20 from Subsection T3.2.3, we find a solution to the original equation (12.4.2.8): y(x)=  CJ 0 (x √ a )ifa > 0, CI 0 (x √ –a )ifa < 0, (12.4.2.10) where J 0 (x) is the Bessel function of the first kind and I 0 (x) is the modified Bessel function of the first kind. In this case, only one solution (12.4.2.10) has been obtained. This is due to the fact that the other solution goes to infinity as x → 0, and hence formula (12.4.2.6) cannot be applied to it; this formula is only valid for finite initial values of the function and its derivatives. 12.4.2-7. Solution of equations using the Laplace integral. Solutions to linear differential equations with polynomial coefficients can sometimes be represented as a Laplace integral in the form y(x)=  K e px u(p) dp.(12.4.2.11) For now, no assumptions are made about the domain of integration K; it could be a segment of the real axis or a curve in the complex plane. Let us exemplify the usage of the Laplace integral (12.4.2.11) by considering equation (12.4.2.7). It follows from (12.4.2.11) that y (k) x (x)=  K e px p k u(p) dp, xy (k) x (x)=  K xe px p k u(p) dp =  e px p k u(p)  K –  K e px d dp  p k u(p)  dp. Substituting these expressions into (12.4.2.7) yields  K e px  n  k=0 a k p k u(p)– n  k=0 b k d dp  p k u(p)   dp + n  k=0 b k  e px p k u(p)  K = 0.(12.4.2.12) This equation is satisfied if the expression in braces vanishes, thus resulting in a linear first-order ordinary differential equation for u(p): u(p) n  k=0 a k p k – d dp  u(p) n  k=0 b k p k  = 0.(12.4.2.13) The remaining term in (12.4.2.12) must also vanish:  n  k=0 b k e px p k u(p)  K = 0.(12.4.2.14) This condition can be met by appropriately selecting the path of integration K. Consider the example below to illustrate the aforesaid. 522 ORDINARY DIFFERENTIAL EQUATIONS Example 2. The linear variable-coefficient second-order equation xy  xx +(x + a + b)y  x + ay = 0 (a > 0, b > 0) (12.4.2.15) is a special case of equation (12.4.2.7) with n = 2, a 2 = 0, a 1 = a + b, a 0 = a, b 2 = b 1 = 1,andb 0 = 0.On substituting these values into (12.4.2.13), we arrive at an equation for u(p): p(p + 1)u  p –[(a + b – 2)p + a – 1]u = 0. Its solution is given by u(p)=p a–1 (p + 1) b–1 . (12.4.2.16) It follows from condition (12.4.2.14), in view of formula (12.4.2.16), that  e px (p + p 2 )u(p)  β α =  e px p a (p + 1) b  β α = 0, (12.4.2.17) where a segment of the real axis, K=[α, β], has been chosen to be the path of integration. Condition (12.4.2.17) is satisfied if we set α =–1 and β = 0. Consequently, one of the solutions to equation (12.4.2.15) has the form y(x)=  0 –1 e px p a–1 (p + 1) b–1 dp. (12.4.2.18) Remark 1. If a is noninteger, it is necessary to separate the real and imaginary parts in (12.4.2.18) to obtain real solutions. Remark 2. By setting α =–∞ and β = 0 in (12.4.2.17), one can find a second solution to equation (12.4.2.15) (at least for x > 0). 12.4.3. Asymptotic Solutions of Linear Equations This subsection presents asymptotic solutions, as ε → 0 (ε > 0), of some higher-order linear ordinary differential equations containing arbitrary functions (sufficiently smooth), with the independent variable being real. 12.4.3-1. Fourth-order linear equations. 1 ◦ . Consider the equation ε 4 y  xxxx – f(x)y = 0 on a closed interval a ≤ x ≤ b. 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(x)] –3/8 exp  – 1 ε  [f(x)] 1/4 dx  , y 2 =[f(x)] –3/8 exp  1 ε  [f(x)] 1/4 dx  , y 3 =[f(x)] –3/8 cos  1 ε  [f(x)] 1/4 dx  , y 4 =[f(x)] –3/8 sin  1 ε  [f(x)] 1/4 dx  . 2 ◦ . Now consider the “biquadratic” equation ε 4 y  xxxx – 2ε 2 g(x)y  xx – f(x)y = 0.(12.4.3.1) Introduce the notation D(x)=[g(x)] 2 + f(x). In the range where the conditions f (x) ≠ 0 and D(x) ≠ 0 are satisfied, the leading terms of the asymptotic expansions of the fundamental system of solutions of equation (12.4.3.1) are described by the formulas y k =[λ k (x)] –1/2 [D(x)] –1/4 exp  1 ε  λ k (x) dx – 1 2  [λ k (x)]  x √ D(x) dx  ; k = 1, 2, 3, 4, where λ 1 (x)=  g(x)+ √ D(x), λ 2 (x)=–  g(x)+ √ D(x), λ 3 (x)=  g(x)– √ D(x), λ 4 (x)=–  g(x)– √ D(x). 12.4. LINEAR EQUATIONS OF ARBITRARY ORDER 523 12.4.3-2. Higher-order linear equations. 1 ◦ . Consider an equation of the form ε n y (n) x – f (x)y = 0 on a closed interval a ≤ x ≤ b. Assume that f ≠ 0. Then the leading terms of the asymptotic expansions of the fundamental system of solutions, as ε → 0,aregivenby y m =  f(x)  – 1 2 + 1 2n exp  ω m ε   f(x)  1 n dx   1 + O(ε)  , where ω 1 , ω 2 , , ω n are roots of the equation ω n = 1: ω m =cos  2πm n  + i sin  2πm n  , m = 1, 2, , n. 2 ◦ . Now consider an equation of the form ε n y (n) x + ε n–1 f n–1 (x)y (n–1) x + ···+ εf 1 (x)y  x + f 0 (x)y = 0 (12.4.3.2) on a closed interval a ≤ x ≤ b.Letλ m = λ m (x)(m = 1, 2, , n) be the roots of the characteristic equation P (x, λ) ≡ λ n + f n–1 (x)λ n–1 + ···+ f 1 (x)λ + f 0 (x)=0. Let all the roots of the characteristic equation be different on the interval a ≤ x ≤ b, i.e., the conditions λ m (x) ≠ λ k (x), m ≠ k, are satisfied, which is equivalent to the fulfillment of the conditions P λ (x, λ m ) ≠ 0. Then the leading terms of the asymptotic expansions of the fundamental system of solutions of equation (12.4.3.2), as ε → 0,aregivenby y m =exp  1 ε  λ m (x) dx – 1 2  [λ m (x)]  x P λλ  x, λ m (x)  P λ  x, λ m (x)  dx  , where P λ (x, λ) ≡ ∂P ∂λ = nλ n–1 +(n – 1)f n–1 (x)λ n–2 + ···+ 2λf 2 (x)+f 1 (x), P λλ (x, λ) ≡ ∂ 2 P ∂λ 2 = n(n – 1)λ n–2 +(n – 1)(n – 2)f n–1 (x)λ n–3 + ···+ 6λf 3 (x)+2f 2 (x). 12.4.4. Collocation Method and Its Convergence 12.4.4-1. Statement of the problem. Approximate solution. 1 ◦ . Consider the linear boundary value problem defined by the equation Ly ≡ y (n) x + f n–1 (x)y (n–1) x + ···+ f 1 (x)y  x + f 0 (x)y = g(x), –1 < x < 1,(12.4.4.1) and the boundary conditions n–1  j=0  α ij y (j) x (–1)+β ij y (j) x (1)  = 0, i = 1, , n.(12.4.4.2) 2 ◦ . We seek an approximate solution to problem (12.4.4.1)–(12.4.4.2) in the form y m (x)=A 1 ϕ 1 (x)+A 2 ϕ 2 (x)+···+ A m ϕ m (x), where ϕ k (x) is a polynomial of degree n + k – 1 that satisfies the boundary conditions (12.4.4.2). The coefficients A k are determined by the linear system of algebraic equations  Ly m – g(x)  x=x i = 0, i = 1, , m,(12.4.4.3) with Chebyshev nodes x i =cos  2i – 1 2m π  , i = 1, , m. 524 ORDINARY DIFFERENTIAL EQUATIONS 12.4.4-2. Convergence theorem for the collocation method. THEOREM. Let the functions f j (x) ( j = 0, , n – 1 )and g(x) be continuous on the interval [–1, 1] and let the boundary value problem (12.4.4.1)–(12.4.4.2) have a unique solution, y(x) . Then there exists an m 0 such that system (12.4.4.3) is uniquely solvable for m ≥ m 0 ; and the limit relations max –1≤x≤1   y (k) m (x)–y (k) (x)   → 0, k = 0, 1, , n – 1;   1 –1   y (n) m (x)–y (n) (x)   2 √ 1 – x 2 dx  1/2 → 0 hold for m →∞ . Remark. A similar result holds true if the nodes are roots of some orthogonal polynomials with some weight function. If the nodes are equidistant, the method diverges. 12.5. Nonlinear Equations of Arbitrary Order 12.5.1. Structure of the General Solution. Cauchy Problem 12.5.1-1. Equations solved for the highest derivative. General solution. An nth-order differential equation solved for the highest derivative has the form y (n) x = f (x, y, y  x , , y (n–1) x ). (12.5.1.1) The general solution of this equation depends on n arbitrary constants C 1 , , C n .In some cases, the general solution can be written in explicit form as y = ϕ(x, C 1 , , C n ). (12.5.1.2) 12.5.1-2. Cauchy problem. The existence and uniqueness theorem. The Cauchy problem: find a solution of equation (12.5.1.1) with the initial conditions y(x 0 )=y 0 , y  x (x 0 )=y (1) 0 , , y (n–1) x (x 0 )=y (n–1) 0 .(12.5.1.3) (At a point x 0 , the values of the unknown function y(x) and all its derivatives of orders ≤ n – 1 are prescribed.) E XISTENCE AND UNIQUENESS THEOREM. Suppose the function f(x, y, z 1 , , z n–1 ) is continuous in all its arguments in a neighborhood of the point (x 0 , y 0 , y (1) 0 , , y (n–1) 0 ) and has bounded derivatives with respect to y , z 1 , , z n–1 in this neighborhood. Then a solution of equation (12.5.1.1) satisfying the initial conditions (12.5.1.3) exists and is unique. 12.5. NONLINEAR EQUATIONS OF ARBITRARY ORDER 525 12.5.1-3. Construction of a differential equation by a given general solution. Suppose a general solution (12.5.1.2) of anunknownnth-order ordinary differential equation is given. The equation corresponding to the general solution can be obtained by eliminating the arbitrary constants C 1 , , C n from the identities y = ϕ(x, C 1 , , C n ), y  x = ϕ  x (x, C 1 , , C n ), y (n) x = ϕ (n) x (x, C 1 , , C n ), obtained by differentiation from formula (12.5.1.2). 12.5.1-4. Reduction of an nth-order equation to a system of n first-order equation. The differential equation (12.5.1.1) is equivalent to the following system of n first-order equations: y  0 = y 1 , y  1 = y 2 , , y  n–2 = y n–1 , y  n–1 = f(x, y 0 , y 1 , , y n–1 ), where the notation y 0 ≡ y is adopted. 12.5.2. Equations Admitting Reduction of Order 12.5.2-1. Equations not containing y, y  x , , y (k) x explicitly. An equation that does not explicitly contain the unknown function and its derivatives up to order k inclusive can generally be written as F  x, y (k+1) x , , y (n) x  = 0 (1 ≤ k + 1 < n). (12.5.2.1) Such equations are invariant under arbitrary translations of the unknown function, y → y + const (the form of such equations is also preserved under the transformation u(x)= y+a k x k +···+a 1 x+a 0 ,wherethea m are arbitrary constants). The substitution z(x)=y (k+1) x reduces (12.5.2.1) to an equation whose order is by k + 1 smaller than that of the original equation, F  x, z, z  x , , z (n–k–1) x  = 0. 12.5.2-2. Equations not containing x explicitly (autonomous equations). An equation that does not explicitly contain x has in the general form F  y, y  x , , y (n) x  = 0.(12.5.2.2) Such equations are invariant under arbitrary translations of the independent variable, x → x + const. The substitution y  x = w(y)(wherey plays the role of the independent variable) reduces by one the order of an autonomous equation. Higher derivatives can be expressed in terms of w and its derivatives with respect to the new independent variable, y  xx = ww  y , y  xxx = w 2 w  yy + w(w  y ) 2 , 526 ORDINARY DIFFERENTIAL EQUATIONS 12.5.2-3. Equations of the form F  ax + by, y  x , , y (n) x  = 0. Such equations are invariant under simultaneous translations of the independent variable and the unknown function, x → x + bc and y → y – ac,wherec is an arbitrary constant. For b = 0, see equation (12.5.2.1). For b ≠ 0, the substitution w(x)=y +(a/b)x leads to an autonomous equation of the form (12.5.2.2). 12.5.2-4. Equations of the form F  x, xy  x – y, y  xx , , y (n) x  = 0. The substitution w(x)=xy  x – y reduces the order of this equation by one. This equation is a special case of the equation F  x, xy  x – my, y (m+1) x , , y (n) x  = 0,wherem = 1, 2, , n – 1.(12.5.2.3) The substitution w(x)=xy  x – my reduces by one the order of equation (12.5.2.3). 12.5.2-5. Homogeneous equations. 1 ◦ . Equations homogeneous in the independent variable are invariant under scaling of the independent variable, x → αx ,where α is an arbitrary constant (α ≠ 0). In general, such equations can be written in the form F  y, xy  x , x 2 y  xx , , x n y (n) x  = 0. The substitution z(y)=xy  x reduces by one the order of this equation. 2 ◦ . Equations homogeneous in the unknown function are invariant under scaling of the unknown function, y → αy,where α is an arbitrary constant (α ≠ 0). Such equations can be written in the general form F  x, y  x /y, y  xx /y, , y (n) x /y  = 0. The substitution z(x)=y  x /y reduces by one the order of this equation. 3 ◦ . Equations 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 constant (α ≠ 0). Such equations can be written in the general form F  y/x, y  x , xy  xx , , x n–1 y (n) x  = 0. The transformation t =ln|x|, w = y/x leads to an autonomous equation considered in Paragraph 12.5.2-2. 12.5.2-6. Generalized homogeneous equations. 1 ◦ . Generalized homogeneous equations (equations homogeneous in the generalized sense) are invariant under simultaneous scaling of the independent variable and the unknown function, x → αx and y → α k y,whereα ≠ 0 is an arbitrary constant and k is a given number. Such equations can be written in the general form F  x –k y, x 1–k y  x , , x n–k y (n) x  = 0. The transformation t =lnx, w = x –k y leads to an autonomous equation considered in Paragraph 12.5.2-2. 12.5. NONLINEAR EQUATIONS OF ARBITRARY ORDER 527 2 ◦ . The most general form of generalized homogeneous equations is F  x n y m , xy  x /y, , x n y (n) x /y  = 0. The transformation z = x n y m , u = xy  x /y reduces the order of this equation by one. 12.5.2-7. Equations of the form F  e λx y n , y  x /y, y  xx /y, , y (n) x /y  = 0. Such equations are invariant under simultaneous translation and scaling of variables, x → x + α and y → βy,whereβ =exp(–αλ/n)and α is an arbitrary constant. The transformation z = e λx y n , w = y  x /y leads to an equation of order n – 1. 12.5.2-8. Equations of the form F  x n e λy , xy  x , x 2 y  xx , , x n y (n) x  = 0. Such equations are invariant under simultaneous scaling and translation of variables, x → αx and y →y+β,whereα =exp(–βλ/n)and β is an arbitrary constant. Thetransformation z = x n e λy , w = xy  x leads to an equation of order n – 1. 12.5.2-9. Other equations. Consider the nonlinear differential equation F  x,L 1 [y], ,L k [y]  = 0,(12.5.2.4) where the L s [y] are linear homogeneous differential forms, L s [y]= n s  m=0 ϕ (s) m (x)y (m) x , s = 1, , k. Let y 0 = y 0 (x) be a common particular solution of the linear equations L s [y 0 ]=0 (s = 1, , k). Then the substitution w = ϕ(x)  y 0 (x)y  x – y  0 (x)y  (12.5.2.5) with an arbitrary function ϕ(x) reduces by one the order of equation (12.5.2.4). Example. Consider the third-order equation xy  xxx = f(xy  x – 2y). It can be represented in the form (12.5.2.4) with k = 2, F(x, u, w)=xu – f(w), L 1 [y]=y  xxx ,L 2 [y]=xy  x – 2y. The linear equations L k [y]=0 are y  xxx = 0, xy  x – 2y = 0. These equations have a common particular solution y 0 = x 2 . Therefore, the substitution w = xy  x – 2y (see formula (12.5.2.5) with ϕ(x)=1/x) leads to a second-order autonomous equation: w  xx = f(w). For the solution of this equation, see Example 1 in Subsection 12.3.2. . infinity as x → 0, and hence formula (12.4.2.6) cannot be applied to it; this formula is only valid for finite initial values of the function and its derivatives. 12.4.2-7. Solution of equations using. conditions f (x) ≠ 0 and D(x) ≠ 0 are satisfied, the leading terms of the asymptotic expansions of the fundamental system of solutions of equation (12.4.3.1) are described by the formulas y k =[λ k (x)] –1/2 [D(x)] –1/4 exp  1 ε  λ k (x). arbitrary constant. For b = 0, see equation (12.5.2.1). For b ≠ 0, the substitution w(x)=y +(a/b)x leads to an autonomous equation of the form (12.5.2.2). 12.5.2-4. Equations of the form F  x, xy  x –