Numerical Solution of Differential Equations

Numerical Solution of Differential EquationsLiz BradleyDepartment of Computer ScienceUniversity of ColoradoBoulder, Colorado, USA 80309-0430c 1998 Revised versionc 2002, Report on Curricula and Teaching CT003-981 Ordinary Differential EquationsA differential equation expresses a set of constraints amongthe derivatives of an unknown function. Here s asimple example:ddtx(t) =ax(t)(1)What this means is that the derivative of the unknown function is equal toatimes the unknown function compactness, the independent variable (t, here) is often omitted and the notationdxdtis often abbreviatedwith a dot, like so: xor with a prime, like so:x . Using these two notations, equation (1) becomes x=axorx =ax, solve a differential equation, you (generally) can t justintegrate both sides. Rather, you have to findsome function that satisfies the constraints expressed in that equation in the example above, some functionx(t) whose derivativex (t) is equal to a constant multiple of the function itself1.

The order of an ODE is the degree of the highest derivative in that equation. x′ = ax is a first-order ODE, x′′′ − tanx′ = 2 is a third-order ODE, and the spring-mass equation above is second order. An nth-order ODE can be transformed into n first-order ODEs (an nth-order “ODE system”), and vice versa2. This

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