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'* = *y*^{2/3}/3,
or the Riccati equation below, have 2.

* __Riccati equation__: The
non-linear equation *y'* = *a* *y*^{2}
+ *b* *y* + *c*; Can be reduced
wlog to *y'* = *y*^{2} + c, whose
solution is *y* = –*ψ*^{–1}*ψ'*,
where *ψ''* = –*c* *ψ*;
The latter is to be solved, a linear second-order equation; (Note: *c* may not be constant!).

- Example: *y'*
= ±(*y*^{2}–*c*), with *c* constant;
The 2-parameter family of solutions reduces to 1-parameter; The solution of *ψ''* = *cψ*
with *c* a constant is *ψ* = *A* exp{–*c*^{1/2}*x*}
+ *B* exp{*c*^{1/2}*x*}, 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 d*x*/d*t* = *x*^{1+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''*(

- u''

- u''

*

*

@

@

@

@

@

@

**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*).**
***

@

@

**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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send feedback and suggestions to bombelli at olemiss.edu – modified 29 nov 2015