Editing Nonlinear system (section)

===Ordinary differential equations===
First order [[ordinary differential equation]]s are often exactly solvable by [[separation of variables]], especially for autonomous equations. For example, the nonlinear equation
:<math>\frac{d u}{d x} = -u^2</math>

has <math>u=\frac{1}{x+C}</math> as a general solution (and also the special solution <math>u = 0,</math> corresponding to the limit of the general solution when ''C'' tends to infinity). The equation is nonlinear because it may be written as
:<math>\frac{du}{d x} + u^2=0</math>

and the left-hand side of the equation is not a linear function of <math>u</math> and its derivatives. Note that if the <math>u^2</math> term were replaced with <math>u</math>, the problem would be linear (the [[exponential decay]] problem).

Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield [[closed-form expression|closed-form]] solutions, though implicit solutions and solutions involving [[nonelementary integral]]s are encountered.

Common methods for the qualitative analysis of nonlinear ordinary differential equations include:

*Examination of any [[conserved quantities]], especially in [[Hamiltonian system]]s
*Examination of dissipative quantities (see [[Lyapunov function]]) analogous to conserved quantities
*Linearization via [[Taylor expansion]]
*Change of variables into something easier to study
*[[Bifurcation theory]]
*[[Perturbation theory|Perturbation]] methods (can be applied to algebraic equations too)
*Existence of solutions of Finite-Duration,<ref>{{cite book |author = Vardia T. Haimo |title = 1985 24th IEEE Conference on Decision and Control |chapter = Finite Time Differential Equations |year = 1985 |pages = 1729–1733 |doi = 10.1109/CDC.1985.268832 |s2cid = 45426376 |chapter-url=https://ieeexplore.ieee.org/document/4048613}}</ref> which can happen under specific conditions for some non-linear ordinary differential equations.