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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)=x_{0}.

If f(t,x) fulfills^{2}dt,X(0)=1. This ODE has the solution
X(t)=1/(1-t), which explodes as t increases up to 1.

dX(t)=f(t,X(t))dt, X(0)=x

If f(t,x) fulfills

- f(t,X(t)) is a Lebesgue measurable function.
- |f(t,x)-f(t,y)|≤K|x-y|, for x,y∈ℝ, t≥0

(Lipschitz-condition), - |f(t,x)|
^{2}≤C(1+x^{2}), for x∈ℝ, t≥0

(Linear growth bound),

Questions: Magnus Wiktorsson

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