REFERENCES FOR CHAPTER 11 451 TABLE 11.6 Areas of application of integral transforms (first in the last column come references to appropriate sections of the current book) Area of application Integral transforms References Evaluation of improper integrals Laplace, Mellin Paragraph 7.2.8-5; Ditkin & Prudnikov (1965) Summation of series Laplace (direct and inverse) Paragraphs 8.1.5-2 and 8.4.4-1 Computation of coefficients of asymptotic expansions Laplace, Mellin Paragraph 7.2.9-1 Linear constant- and variable-coefficient ordinary differential equations Laplace, Mellin, Euler, and others Paragraphs 12.4.1-3 and 12.4.2-6; Ditkin & Prudnikov (1965); Doetsch (1974); E. Kamke (1977); Sveshnikov & Tikhonov (1970); LePage (1980) Systems of linear constant-coefficient ordinary differential equations Laplace Paragraph 12.6.1-4; Doetsch (1974); Ditkin & Prudnikov (1965) Linear equations of mathematical physics Laplace, Fourier, Fourier sine, Hankel, Kontorovich–Lebedev, and others Section 14.5; Doetsch (1974); Ditkin & Prudnikov (1965); Antimirov (1993); Sneddon (1995); Zwillinger (1997); Bracewell (1999); Polyanin (2002); Duffy (2004) Linear integral equations Laplace, Mellin, Fourier, Meler–Fock, Euler, and others Subsections 16.1.3, 16.2.3, 16.3.2, 16.4.6; Krasnov, Kiselev, & Makarenko (1971); Ditkin & Prudnikov (1965); Samko, Kilbas, & Marichev (1993); Polyanin & Manzhirov (1998) Nonlinear integral equations Laplace, Mellin, Fourier Paragraphs 16.5.2-1 and 16.5.3-2; Krasnov, Kiselev, & Makarenko (1971); Polyanin & Manzhirov (1998) Linear difference equations Laplace Ditkin & Prudnikov (1965) Linear differential-difference equations Laplace Bellman & Roth (1984) Linear integro-differential equations Laplace, Fourier Paragraph 11.5.2-2; LePage (1980) References for Chapter 11 Antimirov, M. Ya., Applied Integral Transforms, American Mathematical Society, Providence, Rhode Island, 1993. Bateman, H. and Erd ´ elyi, A., Tables of Integral Transforms. Vols. 1 and 2, McGraw-Hill, New York, 1954. Beerends, R. J., ter Morschem, H. G., and van den Berg, J. C., Fourier and Laplace Transforms, Cambridge University Press, Cambridge, 2003. Bellman,R.andRoth,R.,The Laplace Transform, World Scientific Publishing Co., Singapore, 1984. Bracewell, R., The Fourier Transform and Its Applications, 3rd Edition, McGraw-Hill, New York, 1999. Brychkov, Yu. A. and Prudnikov, A. P., Integral Transforms of Generalized Functions, Gordon & Breach, New York, 1989. Davis, B., Integral Transforms and Their Applications, Springer-Verlag, New York, 1978. Ditkin, V. A. and Prudnikov, A. P., Integral Transforms and Operational Calculus, Pergamon Press, New York, 1965. Doetsch, G., Handbuch der Laplace-Transformation. Theorie der Laplace-Transformation,Birkh ¨ auser, Basel– Stuttgart, 1950. Doetsch, G., Handbuch der Laplace-Transformation. Anwendungen der Laplace-Transformation,Birkh ¨ auser, Basel–Stuttgart, 1956. Doetsch, G., Introduction to the Theory and Application of the Laplace Transformation, Springer-Verlag, Berlin, 1974. Duffy,D.G.,Transform Methods for Solving Partial Differential Equations, 2nd Edition, Chapman & Hall/CRC Press, Boca Raton, 2004. 452 INTEGRAL TRANSFORMS Hirschman, I. I. and Widder, D. V., The Convolution Transform, Princeton University Press, Princeton, New Jersey, 1955. Kamke, E.,Differentialgleichungen: L ¨ osungsmethoden und L ¨ osungen, I, Gew ¨ ohnliche Differentialgleichungen, B. G. Teubner, Leipzig, 1977. Krantz, S. G., Handbook of Complex Variables,Birkh ¨ auser, Boston, 1999. Krasnov, M. L., Kiselev, A. I., and Makarenko, G. I., Problems and Exercises in Integral Equations,Mir Publishers, Moscow, 1971. LePage, W. R., Complex Variables and the Laplace Transform for Engineers, Dover Publications, New York, 1980. Miles, J. W., Integral Transforms in Applied Mathematics, Cambridge University Press, Cambridge, 1971. Oberhettinger, F., Tables of Bessel Transforms, Springer-Verlag, New York, 1972. Oberhettinger, F., Tables of Fourier Transforms and Fourier Transforms of Distributions, Springer-Verlag, Berlin, 1980. Oberhettinger, F., Tables of Mellin Transforms, Springer-Verlag, New York, 1974. Oberhettinger, F. and Badii, L., Tables of Laplace Transforms, Springer-Verlag, New York, 1973. Pinkus, A. and Zafrany, S., Fourier Series and Integral Transforms, Cambridge University Press, Cambridge, 1997. Polyanin, A. D., Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002. Polyanin,A.D.andManzhirov,A.V.,Handbook of Integral Equations, CRC Press, Boca Raton, 1998. Poularikas, A. D., The Transforms and Applications Handbook, 2nd Edition, CRC Press, Boca Raton, 2000. Prudnikov,A.P.,Brychkov,Yu.A.,andMarichev,O.I.,Integrals and Series, Vol. 4, Direct Laplace Transform, Gordon & Breach, New York, 1992. Prudnikov, A. P., Brychkov, Yu. A., and Marichev, O. I., Integrals and Series, Vol. 5, Inverse Laplace Transform, Gordon & Breach, New York, 1992. Samko, S. G., Kilbas, A. A., and Marichev, O. I., Fractional Integrals and Derivatives. Theory and Applica- tions, Gordon & Breach, New York, 1993. Sneddon, I., Fourier Transforms, Dover Publications, New York, 1995. Sneddon, I., The Use of Integral Transforms, McGraw-Hill, New York, 1972. Sveshnikov, A. G. and Tikhonov, A. N., Theory of Functions of a Complex Variable [in Russian], Nauka Publishers, Moscow, 1970. Titchmarsh,E.C.,Introduction to the Theory of Fourier Integrals, 3rd Edition, Chelsea Publishing, New York, 1986. Zwillinger, D., CRC Standard Mathematical Tables and Formulae, 31st Edition, CRC Press, Boca Raton, 2002. Zwillinger, D., Handbook of Differential Equations, 3rd Edition, Academic Press, Boston, 1997. Chapter 12 Ordinary Differential Equations 12.1. First-Order Differential Equations 12.1.1. General Concepts. The Cauchy Problem. Uniqueness and Existence Theorems 12.1.1-1. Equations solved for the derivative. General solution. A first-order ordinary differential equation* solved for the derivative has the form y x = f (x, y). (12.1.1.1) Sometimes it is represented in terms of differentials as dy = f(x, y)dx. A solution of a differential equation is a function y(x) that, when substituted into the equation, turns it into an identity. The general solution of a differential equation is the set of all its solutions. In some cases, the general solution can be represented as a function y = ϕ(x, C) that depends on one arbitrary constant C; specificvaluesofC define specific solutions of the equation (particular solutions). In practice, the general solution more frequently appears in implicit form, Φ(x, y, C)=0, or parametric form, x = x(t, C), y = y(t, C). Geometrically, the general solution (also called the general integral) of an equation is a family of curves in the xy-plane depending on a single parameter C; these curves are called integral curves of the equation. To each particular solution (particular integral) there corresponds a single curve that passes through a given point in the plane. For each point (x, y), the equation y x = f(x, y)defines a value of y x , i.e., the slope of the integral curve that passes through this point. In other words, the equation generates a field of directions in the xy-plane. From the geometrical point of view, the problem of solving a first-order differential equation involves finding the curves, the slopes of which at each point coincide with the direction of the field at this point. Figure 12.1 depicts the tangent to an integral curve at a point (x 0 , y 0 ); the slope of the integral curve at this point is determined by the right-hand side of equation (12.1.1.1): tan α = f(x 0 , y 0 ). The little lines show the field of tangents to the integral curves of the differential equation (12.1.1.1) at other points. 12.1.1-2. Equations integrable by quadrature. To integrate a differential equation in closed form is to represent its solution in the form of formulas written using a predefined bounded set of allowed functions and mathematical operations. A solution is expressed as a quadrature if the set of allowed functions consists of the elementary functions and the functions appearing in the equation and the allowed * In what follows, we often call an ordinary differential equation a “differential equation” or, even shorter, an “equation.” 453 454 ORDINARY DIFFERENTIAL EQUATIONS D α y x x y 0 0 O Figure 12.1. The direction field of a differential equation and the integral curve passing through a point (x 0 , y 0 ). mathematical operations are the arithmetic operations, a finite number of function compo- sitions, and the indefinite integral. An equation is said to be integrable by quadrature if its general solution can be expressed in terms of quadratures. 12.1.1-3. Cauchy problem. The uniqueness and existence theorems. The Cauchy problem: find a solution of equation (12.1.1.1) that satisfies the initial condition y = y 0 at x = x 0 ,(12.1.1.2) where y 0 and x 0 are some numbers. Geometrical meaning of the Cauchy problem: find an integral curve of equation (12.1.1.1) that passes through the point (x 0 , y 0 ); see Fig. 12.1. Condition (12.1.1.2) is alternatively written y(x 0 )=y 0 or y| x=x 0 = y 0 . T HEOREM (EXISTENCE,PEANO). Let the function f(x, y) be continuous in an open domain D of the xy -plane. Then there is at least one integral curve of equation (12.1.1.1) that passes through each point (x 0 , y 0 ) D; each of these curves can be extended at both ends up to the boundary of any closed domain D 0 ⊂ D such that (x 0 , y 0 ) belongs to the interior of D 0 . THEOREM (UNIQUENESS). Let the function f(x, y) be continuous in an open domain D and have in D a bounded partial derivative with respect to y (or the Lipschitz condition holds: |f(x, y)–f(x, z)| ≤ M|y – z|, where M is some positive number). Then there is a unique solution of equation (12.1.1.1) satisfying condition (12.1.1.2). 12.1.1-4. Equations not solved for the derivative. The existence theorem. A first-order differential equation not solved for the derivative can generally be written as F (x, y, y x )=0.(12.1.1.3) T HEOREM (EXISTENCE AND UNIQUENESS). There exists a unique solution y = y(x) of equation (12.1.1.3) satisfying the conditions y| x=x 0 = y 0 and y x | x=x 0 = t 0 , where t 0 is one of the real roots of the equation F (x 0 , y 0 , t 0 )=0 if the following conditions hold in a neighborhood of the point (x 0 , y 0 , t 0 ) : 1. The function F (x, y, t) is continuous in each of the three arguments. 2. The partial derivative F t exists and is nonzero. 3. There is a bounded partial derivative with respect to y , |F y | ≤ M . The solution exists for |x – x 0 | ≤ a, where a is a (sufficiently small) positive number. 12.1. FIRST-ORDER DIFFERENTIAL EQUATIONS 455 12.1.1-5. Singular solutions. 1 ◦ . A point (x, y) at which the uniqueness of the solution to equation (12.1.1.3) is violated is called a singular point. If conditions 1 and 3 of the existence and uniqueness theorem hold, then F (x, y, t)=0, F t (x, y, t)=0 (12.1.1.4) simultaneously at each singular point. Relations (12.1.1.4) define a t-discriminant curve in parametric form. In some cases, the parameter t can be eliminated from (12.1.1.4) to give an equation of this curve in implicit form, Ψ(x, y)=0. If a branch y = ψ(x)ofthecurve Ψ(x, y)=0 consists of singular points and, at the same time, is an integral curve, then this branch is called a singular integral curve and the function y = ψ(x)isasingular solution of equation (12.1.1.3). 2 ◦ . The singular solutions can be found by identifying the envelope of the family of integral curves, Φ(x, y, C)=0, of equation (12.1.1.3). The envelope is part of the C-discriminant curve, which is defined by the equations Φ(x, y, C)=0, Φ C (x, y, C)=0. The branch of the C-discriminant curve at which (a) there exist bounded partial derivatives, |Φ x | < M 1 and |Φ y | < M 2 ,and (b) |Φ x | + |Φ y | ≠ 0 is the envelope. 12.1.1-6. Point transformations. In the general case, a point transformation is defined by x = F (X, Y ), y = G(X, Y ), (12.1.1.5) where X is the new independent variable, Y = Y (X) is the new dependent variable, and F and G are some (prescribed or unknown) functions. The derivative y x under the point transformation (12.1.1.5) is calculated by y x = G X + G Y Y X F X + F Y Y X , where the subscripts X and Y denote the corresponding partial derivatives. Transformation (12.1.1.5) is invertible if F X G Y – F Y G X ≠ 0. Point transformations are used to simplify equations and reduce them to known equa- tions. Sometimes a point transformation allows the reduction of a nonlinear equation to a linear one. Example. The hodograph transformation is an important example of a point transformation. It is defined by x = Y , y = X, which means that y is taken to be the independent variable and x the dependent one. In this case, the derivative is expressed as y x = 1 X Y . Other examples of point transformations can be found in Subsections 12.1.2 and 12.1.4– 12.1.6. 456 ORDINARY DIFFERENTIAL EQUATIONS 12.1.2. Equations Solved for the Derivative. Simplest Techniques of Integration 12.1.2-1. Equations with separated or separable variables. 1 ◦ .Anequation with separated variables (a separated equation)hastheform f(y)y x = g(x). Equivalently, the equation can be rewritten as f (y) dy =g(x) dx (the right-hand side depends on x alone and the left-hand side on y alone). The general solution can be obtained by termwise integration: f(y) dy = g(x) dx + C, where C is an arbitrary constant. 2 ◦ .Anequation with separable variables (a separable equation) is generally represented by f 1 (y)g 1 (x)y x = f 2 (y)g 2 (x). Dividing the equation by f 2 (y)g 1 (x), one obtains a separated equation. Integrating yields f 1 (y) f 2 (y) dy = g 2 (x) g 1 (x) dx + C. Remark. In termwise division of the equation by f 2 (y)g 1 (x), solutions corresponding to f 2 (y)=0 can be lost. 12.1.2-2. Equation of the form y x = f(ax + by). The substitution z = ax + by brings the equation to a separable equation, z x = bf (z)+a; see Paragraph 12.1.2-1. 12.1.2-3. Homogeneous equations y x = f(y/x). 1 ◦ .Ahomogeneous equation remains the same under simultaneous scaling (dilatation) of the independent and dependent variables in accordance with the rule x → αx, y → αy, where α is an arbitrary constant (α ≠ 0). Such equations can be represented in the form y x = f y x . The substitution u = y/x brings a homogeneous equation to a separable one, xu x =f(u)–u; see Paragraph 12.1.2-1. 2 ◦ . The equations of the form y x = f a 1 x + b 1 y + c 1 a 2 x + b 2 y + c 2 can be reduced to a homogeneous equation. To this end, for a 1 x + b 1 y ≠ k(a 2 x + b 2 y), one should use the change of variables ξ = x – x 0 , η = y – y 0 , where the constants x 0 and y 0 are determined by solving the linear algebraic system a 1 x 0 + b 1 y 0 + c 1 = 0, a 2 x 0 + b 2 y 0 + c 2 = 0. 12.1. FIRST-ORDER DIFFERENTIAL EQUATIONS 457 As a result, one arrives at the following equation for η = η(ξ): η ξ = f a 1 ξ + b 1 η a 2 ξ + b 2 η . On dividing the numerator and denominator of the argument of f by ξ, one obtains a homogeneous equation whose right-hand side is dependent on the ratio η/ξ only: η ξ = f a 1 + b 1 η/ξ a 2 + b 2 η/ξ . For a 1 x + b 1 y = k(a 2 x + b 2 y), see the equation of Paragraph 12.1.2-2. 12.1.2-4. Generalized homogeneous equations and equations reducible to them. 1 ◦ .Ageneralized homogeneous equation (a homogeneous equation in the generalized sense) remains the same under simultaneous scaling of the independent and dependent variables in accordance with the rule x → αx, y → α k y,whereα ≠ 0 is an arbitrary constant and k is some number. Such equations can be represented in the form y x = x k–1 f(yx –k ). The substitution u = yx –k brings a generalized homogeneous equation to a separable equation, xu x = f (u)–ku; see Paragraph 12.1.2-1. Example. Consider the equation y x = ax 2 y 4 + by 2 .(12.1.2.1) Let us perform the transformation x = α¯x, y = α k ¯y and then multiply the resulting equation by α 1–k to obtain ¯y ¯x = aα 3(k+1) ¯x 2 ¯y 4 + bα k+1 ¯y 2 .(12.1.2.2) It is apparent that if k =–1, the transformed equation (12.1.2.2) is the same as the original one, up to notation. This means that equation (12.1.2.1) is generalized homogeneous of degree k =–1. Therefore the substitution u = xy brings it to a separable equation: xu x = au 4 + bu 2 + u. 2 ◦ . The equations of the form y x = yf(e λx y) can be reduced to a generalized homogeneous equation. To this end, one should use the change of variable z = e x and set λ =–k. 12.1.2-5. Linear equation y x + f(x)y = g(x). A first-order linear equation is written as y x + f(x)y = g(x). The solution is sought in the product form y = uv,where v = v(x) is any function that satisfies the “truncated” equation v x + f(x)v = 0 [as v(x) one takes the particular solution v = e –F ,whereF = f(x) dx]. As a result, one obtains the following separable equation for u = u(x): v(x)u x = g(x). Integrating it yields the general solution: y(x)=e –F e F g(x) dx + C , F = f(x) dx, where C is an arbitrary constant. . Zafrany, S., Fourier Series and Integral Transforms, Cambridge University Press, Cambridge, 1997. Polyanin, A. D., Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman. Boca Raton, 2002. Polyanin,A.D.andManzhirov,A.V. ,Handbook of Integral Equations, CRC Press, Boca Raton, 1998. Poularikas, A. D., The Transforms and Applications Handbook, 2nd Edition, CRC Press,. 1971. Oberhettinger, F., Tables of Bessel Transforms, Springer-Verlag, New York, 1972. Oberhettinger, F., Tables of Fourier Transforms and Fourier Transforms of Distributions, Springer-Verlag, Berlin,