Chapter D2.1, Problem 59E

a.

To determine

To show: The expression $\frac{d}{dt}\left(y^{\prime }{\left(t\right)}^2\right)=2y^{\prime }\left(t\right){y^{\prime }}^{\prime }\left(t\right)$ by using chain Rule.

b.

To determine

To show: The original differential equation ${\left(y^{\prime }{\left(t\right)}^2\right)}^{\prime }=2$.

c.

To determine

To integrate: The expression ${\left(y^{\prime }{\left(t\right)}^2\right)}^{\prime }=2$ with respect to t to obtain $y^{\prime }\left(t\right)=\pm \sqrt{2t+c_1}$.

d.

To determine

To show: The general solution of the above equation has two families of solutions, $y=c_2+\frac{1}{3}{\left(2t+c_1\right)}^{\frac{3}{2}}\text{ and }y=c_2-\frac{1}{3}{\left(2t+c_1\right)}^{\frac{3}{2}},\text{ where }{c}_1\text{ and }{c}_2\text{ is an arbitrary constant}\text{.}$