Chapter 2.2, Problem 29E

To determine

a)

To show:

The differential equation $y={\left(\frac{2x}{3}+C\right)}^{3/2}$ is the solution of equation $dy/dx=y^{1/3}$.

To determine

b)

To show:

The initial value problem $dy/dx=y^{1/3}$ with $y\left(0\right)=0$ is satisfied for $C=0$ by $y={\left(2x/3\right)}^{3/2}$ for $x\ge 0$.

To determine

c)

To show:

The constant function $y\equiv 0$ also satisfies the initial value problem given in part (b).

To determine

d)

To show:

The conditions of Theorem 1 on page 11 are not satisfied.