## Glossary

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# Differential equation

A differential equation or more precisely an ordinary differential equation (ODE) is an equation concerning the derivative of a function. A first order one dimensional ODE can be written as
dX(t)=f(t,X(t))dt, X(0)=x0.
If f(t,x) fulfills
1. f(t,X(t)) is a Lebesgue measurable function.
2. |f(t,x)-f(t,y)|≤K|x-y|, for x,y∈ℝ, t≥0
(Lipschitz-condition),
3. |f(t,x)|2≤C(1+x2), for x∈ℝ, t≥0
(Linear growth bound),
then there exist a unique global continuous solution to the ODE. The Lipschitz-condition gives uniqueness and Linear growth gives global existence. If the linear growth bound fails then the solution might explode in finite time. Take e.g. dX(t)=X(t)2dt,X(0)=1. This ODE has the solution X(t)=1/(1-t), which explodes as t increases up to 1.