The differential equation given is:

$y' - \frac{1}{x}y = -\frac{2}{x^2} y^{-\frac{5}{3}}$

This is a nonlinear first-order differential equation. To solve this, we could attempt a substitution or a transformation that simplifies the nonlinear terms.

Step-by-step process:

1. Rewrite the equation:

   $y' = \frac{1}{x}y - \frac{2}{x^2} y^{-\frac{5}{3}}$

2. Look for a substitution:

   A good approach is to try a substitution of the form $z = y^{1/3}$, which might simplify the powers of $y$. Since $y = z^3$, we have:

   $y' = 3z^2 \frac{dz}{dx}$

   Substituting $y = z^3$ and $y' = 3z^2 \frac{dz}{dx}$ into the original equation, we get:

   $3z^2 \frac{dz}{dx} = \frac{1}{x} z^3 - \frac{2}{x^2} z^{-\frac{5}{3} \cdot 3}$

   Simplify the powers of $z$:

   $3z^2 \frac{dz}{dx} = \frac{1}{x} z^3 - \frac{2}{x^2} z^{-5}$

3. Divide through by $z^2$:

   $3 \frac{dz}{dx} = \frac{z}{x} - \frac{2}{x^2 z^7}$

4. Simplify further:

   $\frac{dz}{dx} = \frac{z}{3x} - \frac{2}{3x^2 z^7}$

Now, this is still a first-order differential equation in $z$. We could attempt further analysis or an integrating factor, but this substitution has simplified the powers of $y$.

Would you like to continue solving this transformed equation or try a different method?

Related Questions:

1. How do you recognize suitable substitutions for nonlinear differential equations?
2. What are the methods for solving first-order nonlinear differential equations?
3. How does substitution simplify solving complex DEs?
4. What is an integrating factor, and when do you use it?
5. How do you solve first-order separable differential equations?

Tip: Substitution techniques are often key when dealing with nonlinear DEs, transforming them into more familiar forms.