To find the Frobenius series solution for the given second-order ordinary differential equation (ODE):

$x^2 y'' + x y' - 3y = 0,$

we follow these steps:

Step 1: Identify the Regular Singular Point

The equation is in the form of a Frobenius series around $x = 0$, which is a regular singular point since the coefficients $p(x) = \frac{1}{x}$ and $q(x) = -3$ have singularities at $x = 0$.

Step 2: Assume a Frobenius Series Solution

We assume 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 the coefficients.

Step 3: Compute Derivatives

We compute the derivatives of $y(x)$:

$y' = \sum_{n=0}^{\infty} a_n (n+r) x^{n+r-1},$

$y'' = \sum_{n=0}^{\infty} a_n (n+r)(n+r-1) x^{n+r-2}.$

Step 4: Substitute into the ODE

Substituting $y$, $y'$, and $y''$ into the original ODE:

$x^2 \left( \sum_{n=0}^{\infty} a_n (n+r)(n+r-1) x^{n+r-2} \right) + x \left( \sum_{n=0}^{\infty} a_n (n+r) x^{n+r-1} \right) - 3 \left( \sum_{n=0}^{\infty} a_n x^{n+r} \right) = 0.$

This simplifies to:

$\sum_{n=0}^{\infty} a_n (n+r)(n+r-1) x^{n+r} + \sum_{n=0}^{\infty} a_n (n+r) x^{n+r} - 3 \sum_{n=0}^{\infty} a_n x^{n+r} = 0.$

Step 5: Combine Terms

Combining these series gives:

$\sum_{n=0}^{\infty} a_n \left( (n+r)(n+r-1) + (n+r) - 3 \right) x^{n+r} = 0.$

Step 6: Set the Coefficient to Zero

The expression in parentheses must equal zero for all $n$:

$(n+r)(n+r-1) + (n+r) - 3 = 0.$

This simplifies to:

$(n+r)^2 - 3 = 0.$

Step 7: Find the Indicial Roots

Setting $n=0$:

$(r)^2 - 3 = 0 \implies r = \pm \sqrt{3}.$

Step 8: Recurrence Relation

For $n \geq 0$, we have:

$a_n = \frac{3 - (n+r)(n+r-1)}{(n+r)(n+r-1)} a_{n-1}.$

This can be rearranged to get a clearer recurrence relation.

Summary

The solution can be expressed as a power series:

$y(x) = \sum_{n=0}^{\infty} a_n x^{n+r},$

with the recurrence relation determined by the equation derived.

Final Notes

To solve for specific coefficients, you would typically choose $a_0$ or $a_1$ and compute the rest recursively.

Would you like to explore specific coefficients, or do you have any questions about this process? Here are some related topics you might consider:

1. What are the specific values of $a_n$ for $r = \sqrt{3}$?
2. How do you determine the radius of convergence for this series?
3. Can you show how to apply this method to a different ODE?
4. What are other methods to solve second-order ODEs?
5. How does the choice of initial conditions affect the series solution?

Tip: When calculating coefficients, start with small values of $n$ to observe patterns before generalizing the recurrence relation.