Ordinary Differential Equations

First-Order Equations
* Existence theorems: A differential equation y' = g(x,y), with g continuously differentiable in a region R R², admits an infinity of solutions f(x,y,C) = 0, such that for all (x,y) R there passes 1! solution.
* Example: The Langevin equation for brownian motion.
* Non-linear: May have more arbitrary constants than one expects; For example, y' = y^2/3/3, or the Riccati equation below, have 2.
* Riccati equation: The non-linear equation y' = a y² + b y + c; Can be reduced wlog to y' = y² + c, whose solution is y = –^–1', where '' = –c ; The latter is to be solved, a linear second-order equation; (Note: c may not be constant!).
- Example: y' = (y²–c), with c constant; The 2-parameter family of solutions reduces to 1-parameter; The solution of '' = c with c a constant is = A exp{–c^1/2x} + B exp{c^1/2x}, so

- Blow-up in finite time: The solutions of dx/dt = x^{1+}, for > 0.
@ Riccati: Cariñena & Ramos IJMPA(99) [and groups]; Rosu et al JPA(03)mp/01 [generalization]; > s.a. Quaternions.
@ Other types, solutions: Kosovtsov mp/02 [operator method], mp/02 [rational], mp/02 [integrating factors].

Second-Order Equations > s.a. integrable systems; Special Functions; Sturm-Liouville Theory; WKB Method.
* Methods for solution:
- u''(x) + p(x) u'(x) = r(x), substitute v(x):= u'(x);
- u''(x) + p(x) u'(x) + q(x) u(x) = , {see #581};
- u''(x) + p(x) u'(x) + q(x) u(x) = r(x), can be reduced to the form without the q(x) term by u(x) =: v(x) h(x), where h(x) solves the homogeneous equation.
* Eigenvalue problems: –y''(x) + x^2N+2y(x) = x^NE y(x), for – < x < , can be solved in closed form [@ Bender & Wang mp/01].
* Non-linear example: u'' = u², one solution is u = 6/(x+c)².
@ General references: Crampin & Saunders JGP(05) [Cartan theory, duality]; Rafiq et al PLA(08) [homotopy perturbation method].
@ Books: Ayres 52; Coddington & Levinson 55; Nemytskii & Stepanov 60; Pontrjagin 70; Arnold 73, 83; Braun 83; Stroud 74.
@ Eigenvalue problems: Ciftci et al JPA(05)mp/04 [asymptotic iteration method].
@ Delay-differential equations: in Kaplan & Glass 95 [II]; Simmendinger et al PRE(99)mp/01.
@ Non-linear equations: Asch et al mp/01 [h³(h''+h') = 1 as t → ]; Cornejo-Pérez & Rosu PTP(05)mp; Chandrasekar et al JPA(06) [linearization]; Ying & Candès JCP(06) [phase flow method for constructing phase maps].
@ Rational ode's: Avellar et al mp/05, mp/05 [elementary first-order integrals].

Different Types and Other Topics > s.a. differential equations.
* Order reduction: Linear ode's with non-constant coefficients can be reduced in order if one knows any single solution; If u(x) = f(x) is a particular solution, use the ansatz u(x) = f(x) v(x), and the ode becomes an equation of one order less for v'(x).
@ General references: Sakovich PLA(03) [3rd-order, non-linear].
* Linear homogeneous: Can be reduced to an eigenvalue problem.
@ Symmetries: Govinder & Leach JPA(95) [non-local]; Abraham-Shrauner et al JPA(95) [hidden contact symmetries]; Athorne JPA(97) [linear, homogeneous equations].
@ Systems of ode's: Gaeta LMP(97) [normal form]; Mennicken & Moller 03 [boundary eigenvalue problems].
@ Approaches: Diver JPA(93) [genetic algorithm]; Bagarello IJTP(04), same as IJTP(05)?? [non-commutative strategy].

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