9 Initial-Value Problems for Ordinary Differential Equations

Differential equations describe how a quantity changes: the motion of a body, the growth of a population, the decay of a drug in the body. Most of them cannot be solved in closed form, so we compute approximations at a set of points. This section treats the initial-value problem \begin {equation} \label {eq:ivp} \frac {dy}{dt} = f(t,y),\qquad a\leq t\leq b,\qquad y(a) = \alpha . \end {equation} The methods produce approximations \(w_i\approx y(t_i)\) at the mesh points \(t_i = a+ih\), \(i = 0,1,\dots ,N\), with step size \(h = \frac {b-a}{N}\).

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