Chapter 4.3, Problem 31E

Using the mass-spring analogy, predict the behavior as $t\to +\infty$ of the solution to the given initial value problem. Then confirm your prediction by actually solving the problem.

a. $y^{″}+16y=0$; $y\left(0\right)=2$, $y^{\prime }\left(0\right)=0$

b. $y^{″}+100y^{\prime }+y=0$; $y\left(0\right)=1$, $y^{\prime }\left(0\right)=0$

c. $y^{″}-6y^{\prime }+8y=0$; $y\left(0\right)=1$, $y^{\prime }\left(0\right)=0$

d. $y^{″}+2y^{\prime }-3y=0$; $y\left(0\right)=-2$, $y^{\prime }\left(0\right)=0$

e. $y^{″}-y^{\prime }-6y=0$; $y\left(0\right)=1$, $y^{\prime }\left(0\right)=1$