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4. Ordinary Differential Equations (ODE)

Euler's integration scheme: respect it, but don't use it!

Find the solution $y(x)$ of
$\displaystyle L(x,y,y',y'', \dots y^{(n)})$ $\textstyle =$ $\displaystyle 0$  

( $y' \equiv dy/dx$ etc.)

In physics:

- mostly first or second order
- usually given in explicit form, $y'=f(x,y)$ or $y''=g(x,y)$

Second order DE may be written as 2 DEs of first order: $y'=z(x,y)$; $z'=g(x,y)$.



EXAMPLE:
Harmonic oscillator: Instead of $d^{2}x/dt^{2}=-\omega_{0}^{2} x$, write
$\displaystyle \frac{dx}{dt}$ $\textstyle =$ $\displaystyle v \,;\;\;\; \frac{dv}{dt}=-\omega_{0}^{2} x$  

or
$\displaystyle \frac{d \mbox{$\bf y$}}{dt}$ $\textstyle =$ $\displaystyle \mbox{${\bf L}$} \cdot \mbox{$\bf y$}\,,\;\;\;\;{\rm where}\;\;
\...
...\begin{array}{cc}0&1\\  \vspace{-9pt}\\  -\omega_{0}^{2}&0\end{array} \right)$}$  




- If the values of $y$, $y'$ etc. are all given at $x_{0}$:
$\Longrightarrow$Initial Value Problem (IVP).

- If $y$, $y'$ etc. are given at several points $x_{0}, x_{1}, \dots$:
$\Longrightarrow$Boundary Value Problem (BVP).

Typical IVP: equations of motion $d^{2}x/dt^{2}=K/m$; $x(0)$ and $x'(0)$ given

Typical BVP: potential equation $d^{2}\phi/dx^{2}=\rho(x)$; $\phi(x)$ given at boundary points



Subsections
next up previous
Next: 4.1 Initial Value Problems Up: II. Differential Equations Everywhere Previous: II. Differential Equations Everywhere
Franz J. Vesely Oct 2005
See also:
"Computational Physics - An Introduction," Kluwer-Plenum 2001