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Messages - nayan

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Term Test 2 / Re: Problem 2 (main sitting)
« on: November 19, 2019, 01:59:47 PM »

$\text{Using abels theorem,}
W=ce^{-\int p(t)dt}=ce^{-\int 4dt}=ce^{-4t}\\$
$\text{We have the following characteristic polynomial,}\\$
$r=1, r=-2, r=-3\\$
$\therefore y(t)=c_1e^t+c_2e^{-2t}+c_3e^{-3t}\\$
e^t & e^{-2t} & e^{-3t}\\
e^t &  -2e^{-2t} & -3e^{-3t}\\
e^t & 4e^{-2t} & 9e^{-3t}\\
$\text{$\therefore c=-12$ }\\$
$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad 12A=24\\$
$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad A=2$
$\text{We finally have the general solution,}\\ y(t)=c_1e^t+c_2e^{-2t}+c_3e^{-3t}+2te^t$

Term Test 2 / Re: Problem 4 (main sitting)
« on: November 19, 2019, 01:53:29 PM »
Computer-generated sketch:

This doesn't look right. It should instead be an unstable spiral. I've attached an accurate phase portrait.

Yes this appears to be the correct phase portrait.

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