No Constant Solution => No periodic solution

Thread starter cacosomoza
Start date Oct 3, 2010
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Constant Periodic

Oct 3, 2010

cacosomoza

Shall f be continuous function of two real variables. Proof that if equation x''=f(x,x') has not constant solutions, then neither it has periodic solutions.

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Oct 4, 2010

Eynstone

This follows fom the Poincare-Bendixon theorem : a closed orbit must enclose a stationary point.

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...

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