Ordinary Differential Equations

Ordinary Differential Equation

An ordinary differential equation (ODE) is an equation that relates a function of single variable with its derivatives, where denotes an interval. If the highest derivative appeared in the equation is the -th derivative , it is called an -th ODE. The general form of n-th order ODE takes the form A function is called a solution of an -th order ODE if has continuous derivatives up to order such that plugging (t) into the equation gives an identity for each .

e.g. Consider the motion of a spring with one endpoint fixed. let denote its displacement. According to Newton’s second law, where m denotes the mass of the spring. According to Hooke’s law, we have , where is a constant. Then we obtain which is a 2nd order ODE.

Initial Value Problem

An initial value problem for an -th order ODE takes the form

Proposition

Any initial value problem of an -th order ODE can be transformed into an equivalent first order system of ODEs.

Proof Set for all . Then let then we have which is an initial value problem of a first order system of ODEs.

Peano’s Existence Theorem

Let be an open set, let be a continuous function on , and let . Then the initial value problem has a solution defined on for some .