Determine whether the solutions x(t)=0 and x(t)=1 of the single scalar equation $dx/dt=-x(1-x)$ are stable or unstable.
So far in the book I have just done problems like this except with dx/dt=(some matrix)x and then i find the eigenvalues to determine stability, so I am not really sure how to do this problem.
$dx/dt=-x(1-x)$
$=-x+x^2$
Any help would be greatly appreciated
$\endgroup$2 Answers
$\begingroup$Know very little about the topic, but it seems a lot like logistic function:
Hope it leads to a solution.
$\endgroup$ $\begingroup$Hint
Find the lines of stability or equilibrium. Then check to see what sign $dx/dt$ is above and below the line.
For example, one line of stability is $x=0$.
- A little bit above this, say at $x=0.1$, we have a $dx/dt < 0$. A negative slope above $x=0$ indicates that the solution asymptotically approaches $x=0$ as $t\rightarrow\infty$. Stable from above.
- A little bit below it, say at $x=-0.1$, we have $dx/dt >0$. A positive slope below $x=0$ indicates that the solution asymptotically approaches $x=0$ as $t\rightarrow\infty$. Stable from below.
Stable from both above and below $\implies$ $x=0$ is stable.
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