Re: actually something slightly different...

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: : : 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) E^{t^2/2}-t+y₀E^{t^2/2} : as t->infinity : rewrite : y->(Sqrt(Pi)/2 +y₀)E^{t^2/2}-t as t->infinity : t, when compared to E^{t^2/2} at t->infinity is small. Or no : matter what linear coefficient is on E^{t^2/2} (i.e. a), t : will always be smaller than a E^{t^2/2} out at infinity. So : let's drop the small and insignificant t. : y->(Sqrt(Pi)/2 +y₀)E^{t^2/2} : What value of y₀ is the division line between this : coefficient being positive or negative? : y₀=-Sqrt(Pi)/2

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