To solve the differential equation $4x y'' + 2y' + y = 0$ using the Frobenius method, let me outline the steps:

Step-by-Step Solution

1. Rewrite the Differential Equation in Standard Form: Divide the equation by $4x$ to get it in a form that matches the Frobenius method requirement: $y'' + \frac{1}{2x} y' + \frac{1}{4x} y = 0$

2. Assume a Frobenius Series Solution: The Frobenius method assumes a solution of the form: $y(x) = \sum_{n=0}^{\infty} a_n x^{n + r}$ where $r$ is the indicial exponent to be determined, and $a_n$ are coefficients to be found.

3. Find the Indicial Equation: Substitute the series solution into the differential equation, and collect terms of the lowest power of $x$ (which gives us the indicial equation). This step will help us find the value(s) of $r$.

4. Solve for the Coefficients $a_n$: After determining $r$, substitute it back into the series expansion and find a recursive relation for $a_n$.

5. Construct the Solution: Using the values of $r$ and $a_n$, construct the general solution as a series.

Would you like me to go through each calculation in detail?

Related Questions:

1. What is the Frobenius method and when is it used?
2. How do you derive the indicial equation for differential equations?
3. What are regular and irregular singular points in differential equations?
4. Why is it necessary to divide by $4x$ in the initial step?
5. How does the indicial exponent $r$ affect the nature of the solution?

Tip: The Frobenius method is especially useful for solving differential equations around singular points where a Taylor series solution does not converge.