Mathematical Methods for Physicists: A concise introduction - Site Map

ORDINARY DIFFERENTIAL EQUATIONS
Now let us operate G…D† on a product function e t V…t†:
G…D†‰e t V…t†Š ˆ ‰G…D†e t ŠV…t†‡e t ‰G…D†V…t†Š
ˆ e t ‰G…†‡G…D†ŠV…t† ˆe t G…D ‡ †‰V…t†Š:
That is, we have
Rule (b): G…D†‰e t V…t†Š ˆ e t G…D ‡ †‰V…t†Š:
Thus, for example
D 2 ‰e t t 2 Šˆe t …D ‡ † 2 ‰t 2 Š:
Rule (c): G…D 2 † sin kt ˆ G…k 2 † sin kt:
Thus, for example
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D 2 …sin 3t† ˆ1 sin 3t:
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Example 2.12 Damped oscillations (Fig. 2.2)
Suppose we have a spring of natural length L (that is, in its unstretched state). If
we hang a ball of mass m from it and leave the system in equilibrium, the spring
stretches an amount d, so that the ball is now L ‡ d from the suspension point.
We measure the vertical displacement of the ball from this static equilibrium
point. Thus, L ‡ d is y ˆ 0, and y is chosen to be positive in the downward
direction, and negative upward. If we pull down on the ball and then release it,
it oscillates up and down about the equilibrium position. To analyze the oscillation
of the ball, we need to know the forces acting on it:
Figure 2.2.
Damped spring system.
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