Coupled first order equations Quiz

1. $x'=x+2y$, $y'=3x+2y$. Which second order equation does $x$ satisfy?

• A. $x''-3x'+2x=0$
• B. $x''-3x'-4x=0$
• C. $x''+3x'-4x=0$
• D. $x''-3x'+8x=0$

2. $x''-x'-6x=0$ has general solution:

• A. $Ae^{3t}+Be^{-2t}$
• B. $Ae^{-3t}+Be^{2t}$
• C. $Ae^{6t}+Be^{-t}$
• D. $e^{t}(A\cos t+B\sin t)$

3. In a model $y'=2x-y$, the term $2x$ means:

• A. $y$ is killed by $x$
• B. $x$ decreases when $y$ is large
• C. $x$ and $y$ are unrelated
• D. $y$ increases in proportion to $x$

4. $x'=-x-2y$ and $y'=2x-y$ give roots $-1\pm2i$. Describe the populations.

• A. They oscillate with growing amplitude
• B. They grow exponentially
• C. They oscillate, with amplitude decaying to zero
• D. They stay constant

5. After finding $x$, the best way to find $y$ is:

• A. Solve for $y$ with two new constants
• B. Use the first equation to express $y$ in terms of $x$ and $x'$
• C. Set $y=x$
• D. Differentiate $x$ twice