Find the particular component yp of the general solution to

Find the particular component (y_p) of the general solution to y\'\'-2y\'-8y=-1920(t)

Solution

y\'\'-2y\'-8y=-1920(t)

The right hand side (g(t)) is a constant It is a degree 0 (i.e., quadratic) polynomial. Since polynomials, like exponential functions, do not change form after differentiation: the derivative of a polynomial is just another polynomial of one degree less . So Y(t) will, therefore, be a polynomial of the same degree as that of g(t).

Hence y(t)= c

dy/dt= 0, d^2y/dt^2=0. Substitute these in the given equation 0-2*0-8c=-1920

This implies c=1920/8=240. So y(t)=240 is a particular solution.