Question

1 6. The general form of a linear, homogeneous, second-order equation with constant coefficients is dy dy form.

Answer 1

Given that d^2 y/d t^2 + p dy/dt + qy = 0 consider the 2nd order equation with constant coefficient d^2 y/d t^2 + p dy/dt + qy = 0 rightarrow 1 (a) The objective is to write the 1st order system for eq 1 and write this system in matrix form Assume that v = dy/dt du/dt = d^2 y/d t^2 = -p dy/dt - q y from 1 = -pv - qy Therefore The 1^st order system is dy/dt = v dv/dt = -pv - qy dy/dt = 0 + v dv/dt = -qy - pv \r\n The system in matrix form is [dy/dt dv/dt] = (0 -q 1 -p) (y v) dv/dt = Ay where A = (0 -q 1 -p): y = [y v] (b) The objective is to prove that the origin is the only equilibrium point of the system if q notequalto 0 consider the system dy/dt = Ay here A = (0 -q 1 -p) to find the equilibrium point, take