Example 4. Integrate the equation $3 y^{\prime \prime}=y^{-\frac{5}{3}}$.
Solution. This is an equation of the form (10.8), since the right-hand side depends only on $y$. Setting $y^{\prime}=p$, we get $y^{\prime \prime}=p \frac{d p}{d y}$. We rewrite the equation as

$$
3 p \frac{d p}{d y}=y^{-\frac{5}{3}}
$$

Integrating the last equation, we find:

$$
p^{2}=C_{1}-y^{-\frac{2}{3}} \text { or } p= \pm \sqrt{C_{1}-y^{-\frac{2}{3}}}
$$

Since $p=\frac{d y}{d x}$, to determine $y$ we obtain the equation

$$
\pm \frac{d y}{\sqrt{C_{1}-y^{-\frac{2}{3}}}}=d x \text { or } \frac{y^{\frac{1}{3}} d y}{ \pm \sqrt{C_{1} y^{\frac{2}{3}}-1}}=d x
$$

from which

$$
x+C_{2}= \pm \int \frac{y^{\frac{1}{3}} d y}{\sqrt{C_{1} y^{\frac{2}{3}}-1}}
$$

To compute the last integral, we make the substitution

$$
C_{1} y^{\frac{2}{3}}-1=t^{2}
$$

from which

$$
y^{\frac{1}{3}}=\left(t^{2}+1\right)^{\frac{1}{2}} \cdot \frac{1}{C_{1}^{\frac{1}{2}}} ; d y=3 t\left(t^{2}+1\right)^{\frac{1}{2}} \frac{1}{C_{1}^{\frac{3}{2}}} d t
$$

Therefore,

$$
\int \frac{y^{\frac{1}{3}} d y}{\sqrt{C_{1} y^{\frac{2}{3}}-1}}=\frac{1}{C_{1}^{2}} \int \frac{3 t\left(t^{2}+1\right)}{t} d t=\frac{3}{C_{1}^{2}}\left(\frac{t^{3}}{3}+t\right)
$$

Finally, we get

$$
x+C_{2}= \pm \frac{1}{C_{1}^{2}} \sqrt{C_{1} y^{\frac{2}{3}}-1}\left(C_{1} y^{\frac{2}{3}}+2\right)
$$