Transcription of Numerical Solution of Differential Equations
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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.
An ordinary differential equation (ODE) has only one independent variable, and all derivatives in it are taken with respect to that variable. Most often, this variable is time t, but some books use x as the independent variable; pay careful attention to what the derivative is taken with respect to so you don’t get confused.
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