Use educated guessing to find a solution to the overdamped h

Use educated guessing to find a solution to the (over)damped harmonic oscillator system
(dy/dt)=v
(dv/dt)=-3y-3.5v
y(0)=-1.5
v(0)=4.5
The velocity function is v(t)=?
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I used guessing method using y(t)=exp(st)
and obtained a soln v(t)=(-9/2)exp(-2t)+3exp(-3/2t) but im thinking this is my y(t) and am lost on getting v(t).

Solution

A simple spring oscillator is undriven and undamped. The forces governing the motion of the mass are only Newton’s 2nd Law and Hooke’s Law (for springs that obey a linear relation). Putting both together, we obtain the equation for the mass’ displacement with respect to time from the equilibrium position (assuming the system is initially perturbed out of equilibrium to initiate motion). 1. Hooke’s Law: F(y) = ky, where k is the spring constant and y is the displacement from the equilibrium position 2. Newton’s 2nd Law: F(y) = 2 m d y dt2 Equating both: d 2y m = dt2 ky (2.1) Or equivalently: d 2y k = y (2.2) dt2 m The solution to this differential equation is a cosine (or a sine), of frequency k m . That is called the resonant frequency - also called natural frequency or q fundamental frequency - of an undamped spring-like oscillator. Let us call such frequency 0. Hence, 2 0 = k m , and we can solve for k, yielding k = m2 0 . The displacement is given by y(t) = A cos(0t) + C, where A and C are constants corresponding to the initial conditions of the system