October 4th, 2014, 04:21 PM  #1 
Member Joined: May 2012 Posts: 86 Thanks: 0  Diff. Equations problem
I got to prove the following: Suppose f:R>R is continuously differentiable and consider the differential equation x'=f(x) (DE) If f(x)=0 for at least two values of x, then (DE) has at least one unstable equilibrium. (By an unstable equilibrium c it is meant that f(c)=0 and for any epsilon>0 there is some delta>0 such that any solutions that are at a distance less than delta of c will from there onwards remain at a distance less than epsilon from c). However I fail to see how this is always true, since if we let f=0, then DE has only stable equilibriums. Is there something I am not seeing? Answers are appreciated. 
October 4th, 2014, 04:29 PM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,671 Thanks: 2651 Math Focus: Mainly analysis and algebra 
I assume that $f = 0$ is excluded as being the trivial solution. You have written $x^\prime = f(x)$. That suggest that $x$ is itself a function of some other variable. Is that correct? 
October 4th, 2014, 04:37 PM  #3 
Member Joined: May 2012 Posts: 86 Thanks: 0 
Yes, that is correct. However the function defined piecewise by f(x)=0 if x<0, f(x)=x^2 if x>=0 Also appears to have only stable equilibriums, doesn't it? Last edited by Jakarta; October 4th, 2014 at 04:49 PM. 

Tags 
diff, equations, problem 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Is this an ordinary diff eq or a partial diff eq?  Niko Bellic  Calculus  2  July 8th, 2013 10:01 AM 
1st order diff equations  mathkid  Calculus  5  September 10th, 2012 03:10 PM 
diff equations  BonTrust  Calculus  10  January 21st, 2012 05:24 PM 
Help with Diff EQ problem!  thummel1  Calculus  2  September 1st, 2009 05:58 PM 
1st order nonlinear Diff Equations: Any known solution?  Polle  Applied Math  0  December 5th, 2007 09:38 AM 