Differential Equations and Linear Algebra, 4th Edition

7.7 Numerical Methods for Systems

We now discuss the numerical approximation of solutions of systems of differential equations. Our goal is to apply the methods of Sections 2.4 through 2.6 to the initial value problem

$$\mathbf{x}\text{′}=\mathbf{f}(t,\mathbf{x}),\mathbf{x}(t_0)={\mathbf{x}}_0$$ (1)

for a system of m first-order differential equations. In (1) the independent variable is the scalar t, and

$$\mathbf{x}=(x_1,x_2,\dots ,xm)\text{and}\mathbf{f}=(f_1,f_2,\dots ,fm)$$

are vector-valued functions. If the component functions of $\mathbf{f}$ and their first-order partial derivatives are all continuous in a neighborhood of the point $(t_0,{\mathbf{x}}_0)$, then Theorems 3 and 4 of Appendix A guarantee the existence and uniqueness of a solution $\mathbf{x}=\mathbf{x}(t)$ of ...