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.

The query submitted by the user: the DE y' - (1/x)y = (-2/x^2) * y^(-5/3)

Explore how to solve the nonlinear first-order differential equation y' - (1/x)y = (-2/x^2) y^(-5/3) using the substitution method. This step-by-step guide simplifies the problem through the substitution z = y^(1/3), transforming the equation into a more manageable form for further analysis. Suitable for advanced calculus students.

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