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\Large \bf
Numerieke methoden in de natuurwetenschappen\\
\large 
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\noindent 
{\bf Ex.~6: Systems of Ordinary Differential Equations; the anharmonic oscillator}\\ 
Consider the one dimensional motion of a particle subjected to a restoring 
force described by a potential of the form:
\begin{equation}
V(x)=A {{|x|^{B+1}} \over {(B+1)}}
\end{equation}
so that the restoring force is 
\begin{equation}
K(x)=-A |x|^B {x \over |x|}
\end{equation}
The equation of motion is:
\begin{equation}
-M{d^2x \over dt^2} -A |x|^B {x \over |x|}=0
\end{equation}
Take $M$=1kg, A=1Nm$^{-1}$. The initial conditions at $t=0$ are $x$=1m, 
and $dx/dt$=0.

Reduce the equation to a system of two first-order differential equations.
Consider first the harmonic case $B=1$. Calculate the period of vibration 
and compare it to the analytical value of $2\pi\sqrt{M/A}$. Identify the method more suitable to obtain conservation of total (potential+kinetic)
energy  over a few periods of vibration. Explain your considerations in 
achieving this result. 

Next consider other two values of $B$ of your choice and compare 
displacements and velocity with the harmonic case.
 

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