ODELS1

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Write the single $n$ th order equation $u^{(n)}=g(u,u',...,u^{(n-1)},t)\,$ as a first-order system.

Let $x_1=u, x_2=u', ..., x_n=u^{(n-1)}\,$

Now form the first order system

$x_1'=x_2\,$
$x_2'=x_3\,$
$...\,$
$x_{n-1}'=x_n\,$
$x_n'=g(x_1,x_2,...,x_n,t)\,$

With the following initial conditions

$u(t_0)=u, u'(t_0)=u_2, ..., u^{(n-1)}(t_0)=u_n\,$

Let $\vec x = \begin{bmatrix} x_1 \\ x_2 \\ ... \\ x_n \end{bmatrix}\,$

$\vec{x}' = \begin{bmatrix} x_2 \\ x_3 \\ ... \\ g \end{bmatrix}=\vec{f}(x,t)\,$

Let $\vec{f}(x,t)\,$ be continuous for $|t-t_0|\le\alpha, |\vec x - \vec{x}(0)|\le\beta\,$ and satisfy and Lipschitz condition with constant $L\,$ in this region. Let $|\vec{f}(x,t)|\le M\,$ and $\delta=\min\left\{\alpha,\frac{\beta}{M}\right\}\,$ . Then

$\vec{x}(t) = \vec{x}(0)+\int_{t_0}^t f(\vec{x}(s),s)ds\,$ has a unique solution for $|t-t_0|\le\delta\,$ .

Main Page : Ordinary Differential Equations : Linear Systems

ODELS1

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