Please show all steps... d y dy b ky dt2 dt that models the motion of a harmonic

Question

Please show all steps...

d y dy b ky dt2 dt that models the motion of a harmonic oscillator with mass m. spring constant k, and damping coefficient b. In each exercise, we fix two values of these three parameters and obtai a one-parameter family of second-order equ ations. For each one-parameter family. (a) rewrite the one-parameter family as a one-parameter family of linear systems (b) draw the curve in the trace-determinant plane obtained by varying the parameter. and (c) in a brief essay, discuss the different types of behavior exhibited by this one- parameter family.

Explanation / Answer

The general solution of a second order equation contains two arbitrary constants / coefficients. To find a particular solution, therefore, requires two initial values. The initial conditions for a second order equation will appear in the form: y(t0) = y0, and y?(t0) = y?0

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