Question

(d^2x)/(dt^2)-10 dx/dt +34x=0
Explain clearly with how you could use the differential equation to model an undamped oscillator. Then by extending the differential equation into a forced damped oscillator, explain how you could model such an oscillator using a forced second order differential equation.

Answer 1

(d^2x)/(dt^2)-10 dx/dt +34x=0
\[r^2-10r+34=0\\(r-5)^2-25+34=0\\(r-5)^2=-9\\(r-5)=\pm \sqrt{-9}=\pm \sqrt{9} \sqrt{-1}\\r-5=\pm 3i\\r=5 \pm 3i\\so\\x(t) =a e^{(5+3i)t}+be^{(5-3i)t}\\x(t)=e^{5t}(asin(3t)+bcos(3t))=\\e^{5t}(ksin(3t+\theta))\]