The given differential equation is:

\[
x^2 y'' + (2x + 1) y' + 6y = 0
\]

This is a second-order linear differential equation with variable coefficients. To solve it, we can apply methods for solving second-order equations, such as:

1. **Power series solution** (suitable when coefficients are polynomials).
2. **Reduction of order** (if one solution is already known).
3. **Frobenius method** (for singular points).

### Step-by-Step Analysis

The equation has the following form:
\[
x^2 y'' + (2x + 1)y' + 6y = 0
\]

Since it's not immediately obvious how to simplify it using known functions, we would typically look at using a **power series** solution around a point (often \( x = 0 \)) to find the general solution.

### Power Series Solution Method:

We assume that the solution is of the form:
\[
y(x) = \sum_{n=0}^{\infty} a_n x^n
\]
Thus, its derivatives will be:
\[
y'(x) = \sum_{n=1}^{\infty} n a_n x^{n-1}, \quad y''(x) = \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2}
\]

Substitute these into the original differential equation:
\[
x^2 \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2} + (2x + 1) \sum_{n=1}^{\infty} n a_n x^{n-1} + 6 \sum_{n=0}^{\infty} a_n x^n = 0
\]

After substitution, you would combine powers of \(x\) and solve for the coefficients \(a_n\).

Would you like me to proceed further with the series expansion, or would you like to explore another method? Feel free to ask for more details.

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Here are 5 related questions to expand your understanding:

1. How do you determine if a differential equation has a singular point?
2. What is the Frobenius method for solving differential equations?
3. Can you explain how power series solutions are used for variable coefficient equations?
4. What is the difference between ordinary points and singular points in differential equations?
5. How does the method of reduction of order work for second-order differential equations?

**Tip:** When solving differential equations, always check if it's possible to simplify the equation or if special methods like power series or Frobenius are needed at singular points.

This detailed solution explains how to solve the second-order differential equation x^2 y'' + (2x + 1) y' + 6y = 0 using the power series method. Learn the step-by-step approach, including the necessary formulas for derivatives and power series expansions. Ideal for undergraduate students studying differential equations.

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