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MAT244-2013S => MAT244 Math--Tests => Quiz 2 => Topic started by: Victor Ivrii on January 31, 2013, 01:29:59 PM

Title: Day Section, Question 1
Post by: Victor Ivrii on January 31, 2013, 01:29:59 PM
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 $t\to\infty$
$$
y'' - (2\alpha-1)y'+\alpha(\alpha-1)y=0.
$$
Title: Re: Day Section, Question 1
Post by: Brian Bi on January 31, 2013, 04:26:03 PM
The characteristic equation
\begin{equation}
r^2 - (2\alpha-1)r + \alpha(\alpha-1) = 0
\end{equation}
factors as $(r - \alpha)(r - (\alpha - 1))$, so the general solution to the ODE is given by
\begin{equation}
y = A e^{\alpha t} + B e^{(\alpha-1)t}
\end{equation}
where $A, B \in \mathbb{R}$.

We consider the following cases:
We conclude that the answer is: (i) $\alpha < 0$, and (ii) $\alpha > 1$.
Title: Re: Day Section, Question 1
Post by: Zhuolin Liu on February 01, 2013, 04:02:18 PM
I wonder, isn't that when 0<α<1, e(α-1)t tend to 0 as t tend to infinity, and eαt tend to infinity? As a result, isn't that all nonzero solutions become unbounded as t tend to zero when α>0 instead of α>1?
Title: Re: Day Section, Question 1
Post by: Brian Bi on February 01, 2013, 05:43:49 PM
I wonder, isn't that when 0<α<1, e(α-1)t tend to 0 as t tend to infinity, and eαt tend to infinity? As a result, isn't that all nonzero solutions become unbounded as t tend to zero when α>0 instead of α>1?
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 $t \to \infty$.
Title: Re: Day Section, Question 1
Post by: Victor Ivrii on February 01, 2013, 05:51:45 PM
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.