Science:Math Exam Resources/Courses/MATH152/April 2016/Question B 06 (c)

MATH152 April 2016

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[hide] Question B 06 (c)

Consider the three-component differential equation ${\textbf {x}}^{\prime }=A{\textbf {x}}$. The $3\times 3$ matrix $A$ has real entries. It has an eigenvalue $\lambda _{1}=-2$ and an eigenvalue $\lambda _{2}=-1+i$ with corresponding eigenvectors ${\textbf {v}}_{1}=\left[{\begin{array}{c}1\\1\\1\end{array}}\right]$ and ${\textbf {v}}_{2}=\left[{\begin{array}{c}0\\1+i\\1\end{array}}\right].$ (c) Describe all initial conditions for which the solution ${\textbf {x}}(t)$ exhibits oscillatory behaviour. Justify your answer briefly.

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[show] Hint
A solution with oscillatory behavior will have a term involving $\sin t$ or $\cos t$.

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[show] Solution

Recall the general solution as in part (b): ${\textbf {x}}(t)=C_{1}e^{-2t}\left[{\begin{array}{c}1\\1\\1\end{array}}\right]+C_{2}\left[{\begin{array}{c}0\\e^{-t}(\cos t-\sin t)\\e^{-t}\cos t\end{array}}\right]+C_{3}\left[{\begin{array}{c}0\\e^{-t}(\cos t+\sin t)\\e^{-t}\sin t\end{array}}\right].$ If a solution doesn't have oscillatory behaviour, it means that a solution doesn't contain a term having $\cos t$ and $\sin t$. In our case, that happens when $C_{2}=C_{3}=0$. Now, we will relate this condition with the initial data and exclude this case. Suppose the initial condition is ${\textbf {x}}(0)=\left[{\begin{array}{c}a\\b\\c\end{array}}\right].$ Setting $t=0$ in the general solution as in part (b), we get $C_{1}=a,C_{1}+C_{2}+C_{3}=b,$ and $C_{1}+C_{2}=c.$ Then, we can easily see that $C_{2}=C_{3}=0$ is equivalent with $a=b=c=C_{1}$, where $C_{1}$ is an arbitrary number. Therefore, when $a=b=c$, the corresponding initial data doesn't have oscillatory behaviour. In other words, the condition on initial data to make the corresponding solution exhibit oscillatory behavior is that $\color {blue}{\text{at least one of the components differs from the others.}}$