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

  • Thread starter Thread starter Skrew
  • Start date Start date
  • Tags Tags
    First order Odes
Skrew
Messages
131
Reaction score
0
Could someone explain why the graph of a solution can never cross a critical point?
 
Physics news on Phys.org
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.
 
Thread 'Direction Fields and Isoclines'

Thread 'Direction Fields and Isoclines'

I sketched the isoclines for $$ m=-1,0,1,2 $$. Since both $$ \frac{dy}{dx} $$ and $$ D_{y} \frac{dy}{dx} $$ are continuous on the square region R defined by $$ -4\leq x \leq 4, -4 \leq y \leq 4 $$ the existence and uniqueness theorem guarantees that if we pick a point in the interior that lies on an isocline there will be a unique differentiable function (solution) passing through that point. I understand that a solution exists but I unsure how to actually sketch it. For example, consider a...
View full post »

Similar threads

A Existence of unique solutions to a first order ODE on this interval
Replies
2
Views
2K
I Question about solving linear first order non-homogeneous ODEs
Replies
3
Views
3K
I Can the Second Plane Autonomous System Have a Unique Critical Point?
Replies
1
Views
2K
Question about multiple functions for a first order ODE
Replies
1
Views
1K
A Variation of Parameters for System of 1st order ODE
Replies
1
Views
2K
Mathematica - How to Graph Direction Fields of ODEs?
Replies
9
Views
2K
Why does the 2nd order homogeneous linear ODE have 2 general solutions?
Replies
1
Views
2K
MHB How Do I Solve this 2nd Order Non-Linear ODE and Find Its Equilibrium Points?
Replies
3
Views
3K
I Second order non-homogeneous linear ordinary differential equation
Replies
2
Views
3K
An autonomous ODE is simply an ODE
Replies
4
Views
4K

Hot Threads

Recent Insights

Back
Top