Autonomous ODE, determine the limit given an initial condition
For the following autonomous ordinary differential equation, determine the equilibrium points, classify the equilibria as stable or unstable, and find $\lim_{t \rightarrow \infty}x(t)$ under the initial condition $x(0)=2$
$$\dfrac{dx}{dt} = (x-1)(x-2)x(x+1)(x+2)$$
Here is my work:
The equilibrium points are $x=2,1,0,-1,-2$ and $x=-2,0,2$ are unstable, $x= -1,1$ are stable.
What I don't understand is the presence of the initial condition $x(0) =2$. How will the initial condition effect the $\lim_{t \rightarrow \infty}x(t)$ ?
Edit: So if the initial condition is $x(0) =2 $ then $\lim_{t \rightarrow \infty}x(t) = 2$ ??
$\endgroup$ 71 Answer
$\begingroup$You consider a Cauchy problem,
$$\dfrac{dx}{dt} = (x-1)(x-2)x(x+1)(x+2),\quad x(0)=2.$$
By applying Picard's theorem we deduce that there exists a unique solution on some interval $(-\varepsilon,\varepsilon)$. On the other hand, the right hand side is zero for $x=2$, so we can deduce that the solution "$x(t)$ is identically $2$" is a unique solution to our Cauchy problem. Therefore, $\lim x(t)=2$.
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