Why Can't the Graph of a First Order Autonomous ODE Cross a Critical Point?

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SUMMARY

The graph of a solution to a first-order autonomous ordinary differential equation (ODE) cannot cross a critical point due to the nature of the function involved. Specifically, if x' = F(x) and xc is a critical point where F(xc) = 0, then if x(t) = xc at any time t, it follows that x'(t) = F(xc) = 0, indicating that the solution remains at the critical point indefinitely. Furthermore, solutions starting away from critical points can approach them but will never intersect or cross these points.

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Skrew
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Could someone explain why the graph of a solution can never cross a critical point?
 
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OK, to be concrete, let x'=F(x), and let a critical point be xc such that F(xc)=0. Then, if for a given t, x(t)=xc, then x'(t)=F(xc)=0 and, since the equation is first order, it is never going to move again! So: once at a critical point, always at a critical point.

The same argument can be traced "backwards", and that explains why a solution starting away from critical points will never touch one. It may approach, but never touch, much less cross.
 

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