0
$\begingroup$

Consider the following homogeneous linear ODE $$ a\cdot y'' +b\cdot y'+c\cdot y=0$$ where $a$,$b$,$c$ are constants and the characteristic equation $$ a\cdot r^{2}+b\cdot r+c=0$$ has 2 real roots $r_{1}$ and $r_{2}$.

The general solution is $$y=c_{1}\cdot e^{r_{1}t} + c_{2}\cdot e^{r_{2}t}$$

Case 1: $r_{1}>0$,$r_{2}>0$

As $t \rightarrow \infty$, $y \rightarrow +-\infty$, i.e $y$ becomes unbounded.

Case 2: $r_{1}<0$,$r_{2}<0$

As $t \rightarrow \infty$, $y \rightarrow 0$

Case 3: $r_{1}\cdot r_{2}<0$

My intuition is that it is similar to Case 1.

However, $y$ is neither unbounded nor tending to $0$.

$\text{ }$

A) I need a good explanation to understand Case 3.

B) How do we properly describe the behavior of $y$ in Case 3, i.e $r_{1}<0$,$r_{2}>0$ and $r_{1}>0$,$r_{2}<0$?

  • 1
    In Case 3 solutions are mostly unbounded. There are still solutions for which $y \rightarrow 0$: just put $c_i = 0$ for $r_i > 0$. If you convert this 2nd order ODE to the system of two first order ODEs, then you will get a linear system with saddle equilibrium. – 2017-02-24
  • 0
    @Evgeny I understand now. What is the proper description of the behavior of $y$ for the case of $r_{1}<0 , r_{2}>0$ and $r_{1}>0 , r_{2}<0$? Should i compare whether $y\rightarrow 0$ faster or $y\rightarrow \infty$ faster? – 2017-02-25
  • 0
    No, there is no need for comparison: if you add something that goes to infinity to something that goes to zero, the sum still goes to infinity. In terms of equilibrium of linear system your three cases correspond to stable node, to unstable node and to saddle. They are also different in terms of stability: the first one is stable eq., the second one is *completely unstable* (stable when $t \rightarrow -\infty$) and the third one is just unstable. – 2017-02-25
  • 0
    @Evgeny my tutor wrote the solution tends to 0 faster than they tend to +-infinity. Do you know what does this mean? – 2017-03-13
  • 0
    Not really. As I've said, in third case rates almost don't matter: most trajectories are unbounded no matter what. Rates could matter if you were considering how phase volume behaves, but I doubt that this is the case here. And this depends on sum $r_1 + r_2$, so no answer can be given without additional assumptions. – 2017-03-13

0 Answers 0