f(t) = M*(d^2y(t)/dt^2) + b*(dy(t)/dt) + ky(t) Where M = 100 Kg, b = 3000Ns/m, and k = 85kN/m.

Your DE is,

100*(d^2y(t)/dt^2) + 3000*(dy(t)/dt) + 85000y(t) = f(t)

auxiliary equation

100m^2 + 3000m + 85000 = 0

m^2 + 30m + 850 = 0

m^2 + 30m + 15^2 - 225 + 850 = 0

(m+15)^2 = -625

m+15 = +/- 25i

m = -15 - 25i, m = -15 + 25i

The complementary solution then is **y1 = e^(-15){A.cos(25t) + B.sin(25t)}**

This is as far as I can go without knowing what f(t) is.

I need an expression for f(t) in order to compute a particular integral to be added onto the complementary function in order to give a general solution.