Finding the Critical Value of IVP: 'Given DE

[tylerc1991]

Homework Statement

This is more theoretical than anything, but let's say that a problem is stated like:

Consider the IVP: 'given DE'; y(0) = y_0

find the value of y_0 that separates the solutions that grow positively from those that grow negatively as t -> infinity.

The Attempt at a Solution

(1) Is there a name for this particular value of y_0? Is it called the critical value?

(2) is the critical value always found by setting c = 0? (where c is the constant of integration found when solving the IVP)?

Thank you for your help and let me know if I am unclear with anything.

 

[Sethric]

The way I have seen this is:

In a more generalized form in first order (y' = f(x, y)), if there exists a solution that serves as a boundary between two classes of solutions to the differential equation that behave differently, then that solution is called a separatrix. I don't think I have seen a term for the initial value, but it combined with the equation would give you the solution that served as the boundary anyway. Seeing as how picking a different value of x would give a different initial value somewhere along the separatrix, potentially, I think the solution would be more important from a theoretical standpoint.