Chapter 2.3, Problem 8E

Consider any damped harmonic oscillator equation

$m\frac{d^{2}y}{d{t}^2}+b\frac{dy}{dt}+ky=0$
(a) Show that the sum of any two solutions is another solution.
(b) Using the result of part (a), solve the initial-value problem
$\frac{d^{2}y}{d{t}^2}+3\frac{dy}{dt}+2y=0,y(0)=2,v(0)=-3$
(c) Using the result of part (a) in Exercise 7 along with the result of part (a) of this exercise, solve the initial-value problem
$\frac{d^{2}y}{d{t}^2}+3\frac{dy}{dt}+2y=0,y(0)=3,v(0)=-5$
(d) How many solutions to the equation

$\frac{d^{2}y}{d{t}^2}+3\frac{dy}{dt}+2y=0$
do you get if you use the results of Exercise 7 and this exercise along with the guess-and-test method described in this section?