Beverly H. West

The convergence of an Euler approximation of an initial value problem is not always obvious

There is one solution for each b E [0, oo). In particular, there are infinitely many solutions Vb with the same initial condition x (-2) = -1. Since a3...Fxi/ax is unbounded in any region containing the t-axis, it is not surprising that uniqueness of the initial value problem (1) does not hold after the solution hits the t-axis. Let Uh (t) denote the Euler approximation of (1) with stepsize h satisfying Uh (-2) = -1. We investigate what happens as h -* 0. Our main theorem is:

Systems of Nonlinear Differential Equations

The general autonomous differential equation on ℝn is

DiffEq, 3D Views

x\prime = f(x) = \left[ {\begin{array}{*{20}{c}} {{{f}_{1}}(x)} \\ \vdots \\ {{{f}_{n}}(x)} \\ \end{array} } \right],

Qualitative Solution Sketching for First-Order Differential Equations

(1) where f should be thought of as a vector field on an open subset of ℝn. It describes the evolution of innumerable actual systems, and even the two- dimensional

Systems of Differential Equations

\begin{array}{*{20}{c}} {x\prime = f(x,y)} \\ {y\prime = g(x,y)} \\ \end{array}

A New Look at the Airy Equation with Fences and Funnels

(1) case has a great many applications.