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md2xdt2+cdxdt+kx=0m \frac{d^2x}{dt^2} + c \frac{dx}{dt} + kx = 0mdt2d2x+cdtdx+kx=0
c2<4mkc^2 < 4mkc2<4mk
Oscillates with exponentially decaying amplitude
c2=4mkc^2 = 4mkc2=4mk
Returns to equilibrium without oscillating (fastest)
c2>4mkc^2 > 4mkc2>4mk
x(t)=xc+xpx(t) = x_c + x_px(t)=xc+xp
Complementary function
Particular integral
Complex
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