Consider the following homogeneous linear DE: y" + 8a y' + 4 y(t) = 0 where a is a constant. Indicate for what intervals of a the system is undamped, under- damped, critically damped, and over-damped. In each case, (i) give a qualitative description of the oscillations, (ii) give an expression for the roots of the characteristic equation in terms of a – simplify as much as possible, and state if the roots are real, or imaginary (complex). iii) write the general solution for undamped, under-damped, critically damped, and over- damped (in terms of a if necessary).

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