(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

This is the autonomous differential equation: x" - 2x' + 37x = 0

Solve the above DEQ and state whether the critical point (0,0) is stable, unstable, or semi-stable.

2. Relevant equations

Solution to the above DEQ is x = c_{1}e^{x}cos6x + c_{2}e^{x}sin6x

3. The attempt at a solution

I worked out the solution using the quadratic formula and got roots 1[tex]\pm[/tex]6i. This gives you an [tex]\alpha[/tex] of 1 and a [tex]\beta[/tex] of 6, which yields the equation I put in part 2 above.

From there, I read that when you get a general solution in the form x = e^{[tex]\alpha[/tex]t}(c_{1}cos[tex]\beta[/tex]t + c_{2}sin[tex]\beta[/tex]t) with [tex]\alpha[/tex] < 0 and [tex]\beta[/tex] [tex]\neq[/tex]0, then you have a spiral point.

My problem is I'm not sure how to classify the stability of the critical point (0,0).

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# Differential Equation stability at critical point

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