Chapter D1.3, Problem 52E

(a)

To determine

To show: The general solution of the equation $y^{\prime }\left(t\right)=\frac{y\left(y+1\right)}{t\left(t+2\right)}$ can be written as $y\left(t\right)=\frac{\sqrt{t}}{c\sqrt{t+2}-\sqrt{t}}$.

(b)

To determine

To Show: The solution of given initial value problem.is $y\left(t\right)=\frac{\sqrt{t}}{\frac{1+A}{\sqrt{3}A}\sqrt{t+2}-\sqrt{t}}$.

(c)

To determine

To find: The solution $y\left(t\right)=\frac{\sqrt{t}}{\frac{1+A}{\sqrt{3}A}\sqrt{t+2}-\sqrt{t}}$ with initial value $y\left(1\right)=1$ and graph it.

(d)

To determine

To describe: The behavior of the solution of the part (c) as t increases.

(e)

To determine

To find: The solution $y\left(t\right)=\frac{\sqrt{t}}{\frac{1+A}{\sqrt{3}}\sqrt{t+2}-\sqrt{t}}$ with initial value $y\left(1\right)=2$ and graph it.

(f)

To determine

To Describe: The behavior of the solution obtained in part (e) as t increases.

(g)

To determine

To Find: The value of $\underset{t\to \infty }{\lim }y\left(t\right)$ such that the solution is bounded.