The equation is the ideal gas law pV=nRT in disguise for a piston
sliding frictionlessly in a cylinder due to an expanding gas, where
the gas temperature is somehow made to increase linearly with time.
The equation I cited was a prelude to replacing the right-hand side
with some more general variation of the temperature with time:
x*a=f(t)
The idea would be that a piston is held stationary by a pin that is
suddenly removed, so an initial discontinuity in the acceleration
would be expected in fact.
However, as Denker suggests and I considered, there are other
possible ways to start the system, so it would be nice to be more
general.
The scaling argument is cute, but how did you come to choose to
divide x by s^3 and t by s^2? My inclination was instead to scale by
making the variables dimensionless (ie. divide x by x0 and t by t0,
and therefore a by x0/t0^2, where x0 and t0 are some characteristic
length and time) which doesn't work like yours. You appear to have
rigged up yours to give less intercept and slope, is that the
criterion you used to choose them?
In any case, you have convinced me that I have found the asymptotic
solution. Let's define A(t)=sqrt(4k/3)*t^1.5 (A for asymptotic). Can
that help us find the general solution to my original equation
[f(t)=k*t]? I'm thinking of writing something like x(t)=[T(t)+1]*A(t)
where T(t) is a transient that decays to zero as t->infinity. This
approach just leads me to a mess when I sub it into my ODE. But maybe
there's another approach someone can suggest?
--
Carl E Mungan, Assoc Prof of Physics 410-293-6680 (O) -3729 (F)
Naval Academy Stop 9c, 572C Holloway Rd, Annapolis MD 21402-5002 mailto:mungan@usna.eduhttp://usna.edu/Users/physics/mungan/