MATH383

MATH383: A first course in differential equationsApril 7, 2020

Test 3

Solve the following problems (3 course points each). Present a brief motivation of your method of solution. Explicitly state any conditions that must be met for solution procedure to be valid. Organize your computation and writing so the solution you present is readily legible. No credit is awarded for statement of the final answer to a problem without presentation of solution procedure.

This is an open-book test, and you are free to consult the textbook or use software to check your solution. Note however that the questions are so formulated that it is more efficient to draft the solution without use of software or consultation of the textbook; both of those actions would rapidly use up the allotted time. If you studied the course material and understood solutions to the homework assignments, drafting test question solutions in TeXmacs should take about 60 minutes. The allotted time is 3 hours for everyone, thus also providing flexibility for internet connection interruption and special needs accomodation.

Draft your solution in TeXmacs. At least 10 minutes before the submission cut-off time, copy and paste your answer into Sakai. Do not copy and paste these instructions. Select the box of numbered items below, and use menu item Edit->Copy To->TeXmacs to copy to clipboard. Paste into the Sakai answer textbox. When pasted into Sakai your answer will appear between the tags:

<\wide-tabular>
  ...
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Take out your ONYEN card and write your PID here: XXXX-XXX $p q$ . In the following, $p, q$ are the last two digits in your PID

Transform the following system of first-order differential equations
${\begin{cases} y_{1}^{'} = y_{2} \\ y_{2}^{'} = y_{3} \\ y_{3}^{'} = f (t, y_{1}, y_{4}, y_{5}, y_{6}) \\ y_{4}^{'} = y_{5} \\ y_{5}^{'} = y_{4} \\ y_{6}^{'} = g (t, y_{4}, y_{5}, y_{6}) \end{cases} .,$
into a system of two third-order equations for dependent variables $u (t), v (t)$ .
Consider the system of differential equations $𝒚^{'} = 𝑨 𝒚$ ,
$𝒚 = (\begin{array}{l} y_{1} \\ y_{2} \end{array}), 𝑨 = (\begin{array}{ll} a & b \\ - b & 3 a \end{array}) .$
Determine $a, b$ for the fundamental set of solutions to be:
1. ${e^{2 t}, t e^{2 t}}$
2. ${e^{- 6 t} \cos 4 t, e^{- 6 t} \sin 4 t}$
3. ${e^{6 t}, e^{14 t}}$
Complete the set ${e^{(p + q) t}}$ to form a fundamental set of solutions for the system of differential equations $𝒚^{'} = 𝑨 𝒚$
$𝒚 = (\begin{array}{l} y_{1} \\ y_{2} \\ y_{3} \end{array}), 𝑨 = (\begin{array}{lll} p & q & 1 \\ q & p & - 1 \\ 1 & - 1 & 1 \end{array}) .$
Find a fundamental set of solutions for the system of differential equations $𝒚^{'} = 𝑨 𝒚$ ,
$𝒚 = (\begin{array}{l} y_{1} \\ y_{2} \\ y_{3} \end{array}), 𝑨 = (\begin{array}{lll} 1 & - q & - 1 \\ q & 1 & p \\ 1 & - p & 1 \end{array}),$
knowing that as $t \to \infty$ , $e^{- t} 𝒚 (t)$ remains finite and non-zero.