Posted by Brad Paul on October 29, 2002 at 08:21:23:
In Reply to: actually something slightly different... posted by colin on October 28, 2002 at 21:58:05:
: : Am I to understand that your question is:
: : What values of y0 will give finite y as t goes to infinity?
: what I'm looking for is the value of the constant coefficient y0 which is a "critical point" in the family of solutions y(t). The point which separates solutions that behave in two quite different ways- in this case becoming either negatively or positively unbounded.
: I have plotted the function using Mathematica, and can see that this critical value is around approx - 0.8865.
: The book wants us to solve for the "exact value of y0"... and it shows the answer as y0 = - Sqrt(Pi)/2.
: But I am unable to solve the equation y(t) which I've provided for y0. Do you know how to do it?
So we are looking for a critical point at t->infinity. Because I'm a
physicist I'm going to do the math like a physicist.
As t-> infinity, Erf[t]->1 therefore we have
y->Sqrt(Pi)/2 (1) Et^2/2-t+y0Et^2/2
y->(Sqrt(Pi)/2 +y0)Et^2/2-t as t->infinity
t, when compared to Et^2/2 at t->infinity is small. Or no
matter what linear coefficient is on Et^2/2 (i.e. a), t
will always be smaller than a Et^2/2 out at infinity. So
let's drop the small and insignificant t.
What value of y0 is the division line between this
coefficient being positive or negative?
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