You are dealing with a second-order linear differential equation:

\[
x^2 y'' + 3xy' + y = 0
\]

The function \( y = x^{-1} \) is proposed as a solution, so we need to check whether this satisfies the equation.

### Step-by-Step Verification:

1. **Find \( y' \) and \( y'' \) for \( y = x^{-1} \):**

   \[
   y = x^{-1} = \frac{1}{x}
   \]

   \[
   y' = \frac{d}{dx} \left( x^{-1} \right) = -x^{-2}
   \]

   \[
   y'' = \frac{d}{dx} \left( -x^{-2} \right) = 2x^{-3}
   \]

2. **Substitute \( y \), \( y' \), and \( y'' \) into the given equation:**

   \[
   x^2 y'' + 3xy' + y = 0
   \]

   Substituting the values of \( y = x^{-1} \), \( y' = -x^{-2} \), and \( y'' = 2x^{-3} \):

   \[
   x^2 (2x^{-3}) + 3x (-x^{-2}) + x^{-1} = 0
   \]

3. **Simplify each term:**

   \[
   x^2 (2x^{-3}) = 2x^{-1}
   \]

   \[
   3x (-x^{-2}) = -3x^{-1}
   \]

   \[
   x^{-1} = x^{-1}
   \]

4. **Combine all terms:**

   \[
   2x^{-1} - 3x^{-1} + x^{-1} = (2 - 3 + 1) x^{-1} = 0
   \]

   The equation simplifies to \( 0 = 0 \), which is true.

### Conclusion:
The function \( y = x^{-1} \) satisfies the given differential equation.

Would you like more details or have any questions?

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Here are 5 related questions to deepen your understanding:
1. What is the general solution to a second-order linear differential equation?
2. Can this equation have other solutions besides \( y = x^{-1} \)?
3. How would the solution change if there was a non-homogeneous term (e.g., \( y = x^{-1} + \sin(x) \))?
4. How do you solve second-order differential equations using the method of undetermined coefficients?
5. How would you apply variation of parameters to solve such equations?

**Tip:** For verifying solutions to differential equations, always compute the derivatives carefully and substitute them back into the original equation for simplification.

Discover how to solve the second-order linear homogeneous differential equation x^2 y'' + 3xy' + y = 0, and verify that y = x^-1 is a valid solution. This detailed step-by-step solution explores derivative calculations and equation simplifications, ideal for students in undergraduate calculus or differential equations courses.

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