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
of solutions *f*(*x*, *y*, *C*) = 0, such that for all
(*x*, *y*) ∈ \(\mathbb R^2\) 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*; It 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];
Nandakumaran et al 17.

@ __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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