Chapter 3.3, Problem 18P

(a)

To determine

The solution of the initial value problem, $y^{″}+2y^{\prime }+6y=0,y\left(0\right)=2\text{ and }{y}^{\prime }\left(0\right)=\alpha \ge 0$.

(b)

To determine

The value of $\alpha$ such that, $y=0$ when $t=1$.

(c)

To determine

The smallest value of t, in terms of $\alpha$ such that, $t>0$ when $y=0$.

(d)

To determine

The limit of the expression, $\frac{\pi -\arctan \frac{2\sqrt{5}}{\alpha +2}}{\sqrt{5}}$ as $\alpha \to \infty$.