Damped Oscillation & Differential Equations

Question

• Question posed:

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"Particle undergoing damped oscillation by the DE:

x'' + 4x' + 8x = 0

Calculate its displacement x(t) if in addition the initial conditions x(0) = 0 and ˙x(0) = 4 are satisfied."

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• I have obtained the equation:

• How do I calculate the displacement now I have this equation?

Thanks

Answer 1

For homogeneous equation we generally start by assuming general solution with $x=e^{mt} $ . From here we obtain auxilliary equation by putting values of $x'',x',x $ which is $m^2+4m+8=0$ so $m=-2\pm 2i $ so the general solution is $e^{-2t}(a\cos (2t)+b\sin (2t) ) $ a,b are constants. This is obtained using $x=ae^{(-2+2i)t}+be^{(-2-2i)t} $ and some help from Euler formula . Generally three cases for m arise but since you are having more physics background I am not posting it.However if you need it leave a comment.