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Ordinary differential equations

First order ordinary differential equations are often exactly solvable by separation of variables, especially for autonomous equations. For example, the nonlinear equation

has u = 1 x + C {\displaystyle u={\frac {1}{x+C}}} as a general solution (and also the special solution u = 0, {\displaystyle u=0,} corresponding to the limit of the general solution when C tends to infinity). The equation is nonlinear because it may be written as

and the left-hand side of the equation is not a linear function of u {\displaystyle u} and its derivatives. Note that if the u 2 {\displaystyle u^{2}} term were replaced with u {\displaystyle u}, 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 solutions, though implicit solutions and solutions involving nonelementary integrals are encountered.

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