A mass m=1kg, initially at rest and with x=10mm, is connected to a damper with stiffness k=24N/mm and damping constant c=0.2Ns/mm. Given that the differential equation of the system is given by d^2x/dt^2+(dx/dt *c/m)+kx/m=0, find the particular solution.

The system is described by a homogeneous, second order differential equation d²x/dt² +(dx/dt * c/m) + kx/m =0. First, substitute the known constants (m,k,c) to get d²x/dt² +0.2dx/dt + 24x =0. The auxiliary/characteristic equation can then be written as m²+0.2m+24=0, so m=-0.1+4.9i and m=-0.1-4.9i. Since we have both real and imaginary components for m, the general solution will be of the form x(t) = e^pt(Acos(qt) + Bsin(qt)) where p is the real part and q is the imaginary part of m. So, the general solution is x(t) = e^-0.1(Acos(4.9t) + Bsin(4.9t)).

To find the particular solution ,we need to find A and B (thus we need to equations). Notice that the problem states that the mass is initially at rest, which translates to velocity=dx/dt=0 at t=0 (1). The problem also states that initially (t=0), the mass has x=10mm(2). Substitute, t=0 to the general solution, x(t) to find A=10. Then, differentiate x(t) to getdx/dt = -0.1e^-0.1(Acos(4.9t) + Bsin(4.9t))+4.9e^0.1(-Asin(4.9t) + Bcos4.9t)) and substitute t=0 to find B=0.2. Thus the particular solution is:x(t) = e^-0.1(10cos(4.9t) + 0.2sin(4.9t))

NOTE: The wording of this questions is quite tricky, equivalent to an exam style question or even harder! Nonetheless, students are encouraged to practice with these type of questions, to get more intuition into differential equations.