Ordinary Differential Equations  

First-Order Equations
* Existence theorems: A differential equation y' = g(x, y), with g continuously differentiable in a region R ⊂ \({\mathbb R}\)2, admits an infinity of solutions f(x, y, C) = 0, such that for all (x, y) ∈ \({\mathbb R}\) there passes 1! solution.
* Example: The Langevin equation for brownian motion.
* Non-linear: May have more arbitrary constants than one expects; For example, y' = y2/3/3, 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 = –ψ–1ψ', where ψ'' = –c ψ; The latter is to be solved, a linear second-order equation; (Note: c may not be constant!).
- Example: y' = ±(y2c), with c constant; The 2-parameter family of solutions reduces to 1-parameter; The solution of ψ'' = with c a constant is ψ = A exp{–c1/2x} + B exp{c1/2x}, so

\[ \def\ee{{\rm e}}
y = -{\psi'\over\psi} = \sqrt{c}\,{A-B\,\ee^{\pm2\sqrt{c}x}_{\phantom o}\over A+B\,\ee^{\pm2\sqrt{c}x}_{\phantom o}}
= \sqrt{c}\,{1-D\,\ee^{\pm2\sqrt{c}x}_{\phantom o}\over 1+D\,\ee^{\pm2\sqrt{c}x}_{\phantom o}} \;.\]

- Blow-up in finite time: The solutions of dx/dt = x1+ε, for ε > 0.
@ Riccati: Cariñena & Ramos IJMPA(99) [and groups]; Rosu et al JPA(03)mp/01 [generalization]; Cariñena et al EJDE(07)-a0810 [integrable, geometric approach]; > 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) + x2N+2y(x) = xNE y(x), for –∞ < x < ∞, can be solved in closed form [@ Bender & Wang JPA(01)mp].
* 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 equations: Avellar et al AMC(07)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).
* Linear homogeneous: Can be reduced to an eigenvalue problem.
@ General references: Sakovich PLA(03) [third-order, non-linear]; Górka et al CQG(12)-a1208 [with infinitely many derivatives, initial-value problem]; Gurappa et al a1205 [linear].
@ Systems: Gaeta LMP(97) [normal form]; Mennicken & Möller 03 [boundary eigenvalue problems]; Cariñena & de Lucas DissM(11)-a1103 [Lie systems].

References
@ General: Polyanin & Zaitsev 02 [exact solutions, handbook]; Schroers 11 [practical guide]; Deng 14 [lectures, problems and solutions].
@ Symmetries: Govinder & Leach JPA(95) [non-local]; Abraham-Shrauner et al JPA(95) [hidden contact symmetries]; Athorne JPA(97) [linear, homogeneous equations].
@ Approaches: Diver JPA(93) [genetic algorithm]; Bagarello IJTP(04), same as IJTP(05)?? [non-commutative strategy]; Bervillier JPA(09)-a0812 [conformal mappings and other methods]; White 10 [asymptotic analysis].


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