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Reduction of Higher Order ODEs to a System of First Order ODEs

Higher-order ordinary differential equations can usually be re-written as a system of first-order differential equations. If the higher-order ODE can be solved for its largest derivative:

$$\ensuremath{\frac{d^{n}{y}}{d{t}^{n}}} = F(\ensuremath{\frac{d^{n... ...ath{\frac{d^{n-2}{y}}{d{t}^{n-2}}}, \ldots , \ensuremath{\frac{d{y}}{d{t}}}, t)$$ (24-6)

then $n-1$ ``new'' functions can be introduced via

$$\begin{split}y_0(t) & \equiv y(t)\ y_1(t) \equiv & \ensurema... ...d{t}}}, t) = \ensuremath{\frac{d{y_{n-1}}}{d{t}}}\ \end{split}$$ (24-7)

or

$$\ensuremath{\frac{d{}}{d{t}}} \left( \begin{array}{l} y_0\ y_1\\... ...ots\ y_{n-1}\ F(y_{n-1}, y_{n-2}, \ldots , y_1, y_0, t)\ \end{array} \right)$$ (24-8)

For example, the damped harmonic oscillator, $M \ddot{y} + \eta l_o \dot{y} + K_s y = 0$ , can be re-written by introducing the momentum variable, $p = M v = M \dot{y}$ , as the system:

$$\begin{split}\ensuremath{\frac{d{y}}{d{t}}} & = \frac{p}{M}\\... ...remath{\frac{d{p}}{d{t}}} & = -K_s y - \eta l_o p\ \end{split}$$ (24-9)

which has only one critical point $y=p=0$ .

The equation for a free pendulum, $M R^2 \ddot{\theta} + M g R \sin(\theta) = 0$ , can be re-written by introducing the angular momentum variable, $\omega = M R \dot{\theta}$ as the system,

$$\begin{split}\ensuremath{\frac{d{\theta}}{d{t}}} &= \frac{\om... ...th{\frac{d{\omega}}{d{t}}} &= - M g \sin(\theta) \ \end{split}$$ (24-10)

which has two different kinds of critical points: ( $\omega = 0, \theta = n_{even} \pi$ ) and ( $\omega = 0, \theta = n_{odd} \pi$ ).

Finally, the beam equation $EI \ensuremath{\frac{d^{4}{y}}{d{x}^{4}}} = w(x)$ can be rewritten as the system:

$$\ensuremath{\frac{d{}}{d{x}}} \left( \begin{array}{l} y\ m_{slop... ...\left( \begin{array}{l} m_{slope}\ \frac{M}{EI}\ S\ w(x) \end{array} \right)$$ (24-11)

where $m_{slope}$ is the slope of the beam, $M$ is the local bending moment in the beam, $S$ is the local shearing force in the beam, and $w(x)$ is the load density.

This beam equation does not have any interesting critical points.