Franz J. Vesely > CompPhys Tutorial > Differential Equations 

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Part II: Ch. 4

4 Ordinary Differential Equations (ODE)

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


Basic 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: $y'=z(x,y)$; $z'=g(x,y)$.

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} $

$ \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

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

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vesely 2005-10-10

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