| Franz J. Vesely > CompPhys Tutorial > Differential Equations |
4 Ordinary Differential Equations (ODE)
DefinitionsBasic problem: Find the solution $y(x)$ of
$
L(x,y,y',y'', \dots y^{(n)}) = 0
$
(where $y' \equiv dy/dx$ 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
$
\begin{eqnarray}
\frac{\textstyle dx}{\textstyle dt}& = & v ; \;\;\;
\frac{\textstyle dv}{\textstyle dt}=-\omega_{0}^{2} x
\end{eqnarray}
$
or
$
\begin{eqnarray}
\frac{\textstyle d y}{\textstyle dt}&=& L \cdot y , \;\;\;\;
\rm where\;\;
y \equiv \left( \begin{array}{r} x \\ \\ v \end{array} \right) \;\;\; {\rm and}\;\;
L= \left( \begin{array}{cc}0&1\\ \\ -\omega_{0}^{2}&0 \end{array} \right)
\end{eqnarray}
$
- 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
Go to: vesely 2005-10-10
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