Chapter 2.2, Problem 30E

As stated in this section, the separation of equation $\left(2\right)$ on page 42 requires division by $p\left(y\right)$, and this may disguise the fact that the roots of the equation $p\left(y\right)=0$ are actually constant solutions to the differential equation.

a. To explore this further, separate the equation

$\frac{dy}{dx}=\left(x-3\right){\left(y+1\right)}^{2/3}$

to derive the solution,

$y=-1+{\left(x^2/6-x+C\right)}^3$.

b. Show that $y\equiv -1$ satisfies the original equation

$dy/dx=\left(x-3\right){\left(y+1\right)}^{2/3}$.

c. Show that there is no choice of the constant $C$ that will make the solution in part (a) yield the solution $y\equiv -1$. Thus, we lost the solution $y\equiv -1$ when we divided by ${\left(y+1\right)}^{2/3}$.