MAT244-2013S > Quiz 2

Day Section, Question 1

(1/1)

Victor Ivrii:
Determine the values of $\alpha$ , if any, for which all solutions of the following ODE tend to zero as $t\to\infty$ as well as all values of $\alpha$ , if any, for which all nonzero solutions become unbounded as $t\to\infty$
$$y'' - (2\alpha-1)y'+\alpha(\alpha-1)y=0.$$

Brian Bi:
The characteristic equation

r^2 - (2\alpha-1)r + \alpha(\alpha-1) = 0

factors as $(r - \alpha)(r - (\alpha - 1))$, so the general solution to the ODE is given by

y = A e^{\alpha t} + B e^{(\alpha-1)t}

where $A, B \in \mathbb{R}$.

We consider the following cases:

* $\alpha < 0$: Both exponentials will be decaying, so each solution tends to zero as $t \to \infty$.
* $\alpha = 0$ or $\alpha = 1$: Each $y = c$ for constant $c$ is a solution, so there exist solutions that neither tend to zero nor become unbounded as $t \to \infty$.
* $0 < \alpha < 1$: One exponential is growing and the other decaying, so there exist nonzero solutions that tend to zero as well as solutions that tend to infinity.
* $\alpha > 1$: Both exponentials will be growing. The larger of the two, $Ae^{\alpha t}$, dominates as $t \to \infty$, so $y$ is unbounded unless $A = 0$. If $A$ vanishes identically, then all nonzero solutions $Be^{(\alpha-1)t}$ again become unbounded.We conclude that the answer is: (i) $\alpha < 0$, and (ii) $\alpha > 1$.

Zhuolin Liu:
I wonder, isn't that when 0<Î±<1, e(Î±-1)t tend to 0 as t tend to infinity, and eÎ±t tend to infinity? As a result, isn't that all nonzero solutions become unbounded as t tend to zero when Î±>0 instead of Î±>1?

Brian Bi:

--- Quote from: Zhuolin Liu on February 01, 2013, 04:02:18 PM ---I wonder, isn't that when 0<Î±<1, e(Î±-1)t tend to 0 as t tend to infinity, and eÎ±t tend to infinity? As a result, isn't that all nonzero solutions become unbounded as t tend to zero when Î±>0 instead of Î±>1?

--- End quote ---
The coefficient on $e^{\alpha t}$ might be zero, so you can have solutions that are just $A e^{(\alpha-1)t}$. These will decay to zero as $t \to \infty$.

Victor Ivrii:
We will discuss these things in details later but just simple observations:

If we have two characteristic roots $\lambda_2 >0>\lambda_1$ then almost all solutions (with $C_1\ne 0$ and $C_2\ne 0$) are unbounded as $t\to \pm \infty$, solutions $C_2e^{\lambda_2t}$ ($C_2\ne 0$) are unbounded as $t\to +\infty$ and tend to $0$ as $t\to-\infty$ and solutions $C_1e^{\lambda_1t}$ ($C_2\ne 0$) are unbounded as $t\to -\infty$ and tend to $0$ as $t\to +\infty$.

PS.  I prefer to write $+\infty$ rather than $\infty$ to avoid any ambiguity.

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