Science:Math Exam Resources/Courses/MATH152/April 2012/Question 08 (c)

MATH152 April 2012

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Question 08 (c)
Consider the 2 x 2 system of differential equations $\mathbf {x} '(t)={\begin{bmatrix}-1&-2\\2&-1\end{bmatrix}}\mathbf {x} (t)$ where $\mathbf {x} (t)={\begin{bmatrix}x_{1}(t)\\x_{2}(t)\end{bmatrix}}$ is the vector of unknown functions. Based on the results in (a) and (b), write down the general solution in real-valued form.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Remember that $\displaystyle e^{ix}=\cos(x)+i\sin(x)$ for any x. You can use this to decompose the solution into its real and imaginary parts. Since no imaginary component can be expressed in terms of the real components, the real and imaginary components can be used as a basis for the general solution.

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. The complex solution ${\vec {\mathbf {z} }}(t)$ that we found in (a) and (b) is ${\begin{aligned}{\vec {\mathbf {z} }}(t)&=e^{-t}e^{2ti}{\begin{pmatrix}1\\-i\end{pmatrix}}\\&=e^{-t}(\cos 2t+i\sin 2t){\begin{pmatrix}1\\-i\end{pmatrix}}\\&=e^{-t}{\begin{pmatrix}\cos 2t\\\sin 2t\end{pmatrix}}+ie^{-t}{\begin{pmatrix}\sin 2t\\-\cos 2t\end{pmatrix}}\end{aligned}}$ From this we can read off the real-valued form as ${\vec {\mathbf {x} }}(t)=C_{1}e^{-t}{\begin{pmatrix}\cos 2t\\\sin 2t\end{pmatrix}}+C_{2}e^{-t}{\begin{pmatrix}\sin 2t\\-\cos 2t\end{pmatrix}}$