Laplace s Method for Ordinary Differential Equations

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1 Physics 2400 Spring 2016 Laplace s Method for Ordinary Differential Equations Lecture notes by M. G. Rozman Last modified: March 31, 2016 We can [1, pp ], [2, h. VIII], [3, Apps. a-b], [4, h. 18], [5, h. 8.A] represent solutions of some differential equations by definite integrals in which the independent variable appears as a parameter under the integral sign. In this compact form various properties of different solutions to an equation become quite clear, asymptotic expansions can be obtained directly, and numerical computation may be facilitated. One of the most important applications of this method is due to Laplace and affects the equation (a 0 + b 0 x) dn y dx n + (a 1 + b 1 x) dn 1 y dx n (a n + b n x) y = 0, (1) whose coefficients are at most of the first degree in x. Let us try to find a solution of this equation by taking for y an expression of the form y(x) = Z(t) e xt dt (2) where Z is a function of the variable t and where is an unspecified yet integration contour independent of x. We have, d p y dx p = Z(t) t p e xt dt, (3) and, replacing y and its derivatives in the left-hand side of Eq. (1) by Eq. (3), we find Z(t) e xt (P(t) + x Q(t)) dt = 0, (4) where we have set, for brevity, P(t) = a 0 t n + a 1 t n a n, (5) Q(t) = b 0 t n + b 1 t n b n. (6) Page 1 of

2 Integrating Eq. (4) by parts, we get 0 = Z(t) [P(t) + x Q(t)] e xt dt (7) = = Z(t) P(t) e xt dt + Z(t) Q(t) de xt (8) [P(t)Z(t) ddt ] [Q(t)Z(t)] e xt dt + [ Q(t)Z(t) e xt] 2 where the second term in Eq. (9) is evaluated at the end points of the contour. If we choose the contour so as to make this contribution vanish, [ ] Q(t)Z(t) e xt 2 = 0, (10) then Eq. (2) will represent a solution to Eq. (1) if the function Z(t) satisfies the differential equation 1 d [Q(t)Z(t)] P(t)Z(t) = 0. (11) dt Eq. (11) is the first order linear ordinary differential equation that can be solved separating variables: d [Q(t)Z(t)] = P(t)Z(t)dt (12) d [Q(t)Z(t)] Q(t)Z(t) 1 (9) = P(t) dt (13) Q(t) d ln (Q(t)Z(t)) = P(t) dt (14) Q(t) P(t) ln (Q(t)Z(t)) = Q(t) dt + c 1, (15) where c 1 is an integration constant. Exponentiating, ( ) P(t) Q(t)Z(t) = c exp Q(t) dt, (16) where c = exp(c 1 ) is another integration constant; Z(t) = c Q(t) exp ( ) P(t) Q(t) dt. (17) Using Eq. (17) it is possible to determine suitable integration contour(s) to fulfill the requirement of Eq. (10). Page 2 of

3 Integral representation of a solution to a boundary-value problem Example 3 in [6, h. 6] The equation x d3 y + 2y = 0, y(0) = 1, y( ) = 0, (18) dx3 is of Laplace s type. Following the general method, we form the functions P(t) and Q(t): a 0 = 0, b 0 = 1, a 1 = 0, b 1 = 0, a 2 = 0, b 2 = 0, a 3 = 2, b 3 = 0, (19) and P(t) = a 0 t 3 + a 1 t 2 + a 2 t + a 3 = 2, Q(t) = b 0 t 3 + b 1 t 2 + b 2 t + b 3 = t 3, (20) P(t) dt Q(t) dt = 2 t 3 = 1 t 2, (21) Z = 1 t 3 exp ( 1 t 2 ) (22) The definite integral y(x) = e xt 1 t 2 is therefore a particular integral of Eq. (18) if the function takes on the same values at the extremities of the path of integration. t 3 dt (23) Q(t)Z(t) = e xt 1 t 2 (24) Let s assume that x > 0 and choose the integration contour along the negative real axis, < t 0. Q( )Z( ) = Q(0)Z(0) = 0, (25) y(x) 0 e xt 1 t 2 d 1 t 2 = where we changed the integration variables u = 1 t 2, 0 u <. 0 e x u u du, (26) The Airy function The equation d 2 y dx 2 x y = 0 (27) is of Laplace s type. Following the general method, we form the functions a 0 = 1, b 0 = 0, a 1 = 0, b 1 = 0, a 2 = 0, b 2 = 1, (28) Page 3 of

4 P(t) = a 0 t 2 + a 1 t + a 2 = t 2, Q(t) = b 0 t 2 + b 1 t + b 2 = 1, (29) P(t) Q(t) dt = t3 3, t3 Z(t) = e 3, (30) so that the solution can be represented in the form y(x) = e xt t3 3 dt. (31) The path of integration must be chosen so that the function e xt t3 3 (32) vanishes at both ends of it. These ends must therefore go to infinity in the regions of the complex plane of t in which Re t 3 > 0. Hermite Polynomials As an illustration of Laplace s method, we consider Hermite s differential equation, d 2 y dy 2x + 2ν y = 0, (33) dx2 dx where ν is an arbitrary constant. In this case Eq. (17) gives a 0 = 1, a 1 = 0, a 2 = 2ν, (34) b 0 = 0, b 1 = 2, b 2 = 0, (35) P(t) = t 2 + 2ν, (36) Q(t) = 2t, (37) P(t) ( Q(t) dt = t 2 ν ) dt = t2 t 4 ν log t = t2 4 + log 1 t ν. (38) Z(t) = A t2 e tν+1 This may be substituted into Eq. (2) to obtain an integral representation. y(x) = A 4. (39) e xt t2 4 dt (40) tν+1 where A is an arbitrary constant, and the path of integration must be chosen so that e xt t2 2 4 t ν+1 = 0, (41) 1 where the expression is evaluated at the end points of contour. We will discuss the choice of the integration contour when we talk about harmonic oscillator in quantum mechanics. Page 4 of

5 Bessel functions Bessel functions satisfy the equation d 2 f dx d f x dx + ) (1 ν2 x 2 f = 0, (42) where ν is a given constant. The solution is most easily effected if we remove the singularity at x = 0; this may be achieved by the substitution to obtain the equation We have here and consequently The definite integral f (x) = x ν y(x), (43) d f dx = νxν 1 y + x ν dy dx (44) d 2 f dx 2 = ν(ν 1)xν 2 ν 1 dy y + 2νx dx + xν d2 y dx 2 (45) x d2 y dy + (2ν + 1) dx2 dx + x y = 0. (46) P(t) = (2ν + 1) t, Q(t) = 1 + t 2, (47) ( P(t) t dt dt = (2ν + 1) Q(t) 1 + t 2 = ν + 1 ) d(1 + t 2 ) t 2 = ln ( 1 + t 2) 1 ν+ 2 (48) y(x) = Z = ( 1 + t 2) ν 1 2 is therefore a particular integral of the equation (46) if the function (49) e xt ( 1 + t 2) ν 1 2 dt (50) e xt ( 1 + t 2) ν+ 1 2 (51) takes on the same values at the extremities of the path of integration. Laguerre polynomials A differential equation that arises in the study of the hydrogen atom is the Laguerre equation: x f (x) + (1 x) f (x) + λ f (x) = 0. (52) Let us attack the solution to this equation using Laplace s method. Page 5 of

6 References [1] Éduard Goursat. Differential Equations, volume II of A ourse in Mathematical Analysis. Dover Publications, [2] Edward Lindsay Ince. Ordinary Differential Equations. Dover Publications, [3] Lev D. Landau and Evgeny M. Lifshitz. Quantum Mechanics Non-Relativistic Theory, volume III of ourse of Theoretical Physics. Butterworth-Heinemann, 3 edition, [4] B. Brian Davies. Integral transforms and their applications, volume 41 of Texts in Applied Mathematics. Springer Verlag, 3 edition, [5] Hung heng. Advanced Analytic Methods in Applied Mathematics, Science, and Engineering. LuBan Press, [6] arl M. Bender and Steven A. Orszag. Advanced Mathematical Methods for Scientists and Engineers. Springer Verlag, Page 6 of