To solve this differential equation using a power series, let's go through the steps. The differential equation given is:

$y'' - xy' = 0$

We will assume a power series solution of the form: $y(x) = \sum_{n=0}^{\infty} a_n x^n$

Step 1: Compute the Derivatives

1. The first derivative $y'$ is: $y' = \sum_{n=1}^{\infty} n a_n x^{n-1}$

2. The second derivative $y''$ is: $y'' = \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2}$

Step 2: Substitute into the Differential Equation

Substitute $y$, $y'$, and $y''$ into the equation $y'' - xy' = 0$:

$\sum_{n=2}^{\infty} n(n-1) a_n x^{n-2} - x \sum_{n=1}^{\infty} n a_n x^{n-1} = 0$

Rewrite the second term by distributing $x$ inside the summation: $\sum_{n=2}^{\infty} n(n-1) a_n x^{n-2} - \sum_{n=1}^{\infty} n a_n x^n = 0$

Step 3: Align the Series Indices

To combine terms, we need the powers of $x$ to match. Let’s re-index the first summation so both summations start from the same power of $x$:

1. In the first series, let $k = n - 2$, so $n = k + 2$: $\sum_{k=0}^{\infty} (k+2)(k+1) a_{k+2} x^k$

2. In the second series, let $k = n - 1$, so $n = k + 1$: $\sum_{k=0}^{\infty} (k+1) a_{k+1} x^k$

Step 4: Set Up the Recurrence Relation

Now we have: $\sum_{k=0}^{\infty} \left[ (k+2)(k+1) a_{k+2} - (k+1) a_{k+1} \right] x^k = 0$

For this series to be zero for all $x$, each coefficient of $x^k$ must individually be zero: $(k+2)(k+1) a_{k+2} - (k+1) a_{k+1} = 0$

Step 5: Solve the Recurrence Relation

This simplifies to: $a_{k+2} = \frac{a_{k+1}}{k+2}$

Using this recurrence relation, we can generate the coefficients in terms of $a_0$ and $a_1$.

Step 6: Determine the Interval of Convergence

To determine the interval of convergence, we generally rely on the radius of convergence for the power series, which in this case can be found using the ratio test. However, without specific initial conditions or further constraints, this differential equation typically has a power series that converges for all $x$ (entire function).

Would you like more detailed steps on computing individual terms or an exploration of specific examples?

Follow-up Questions:

1. What is the radius of convergence in general for a power series solution to a differential equation?
2. How does the form of a recurrence relation impact the convergence of a power series?
3. How do initial conditions influence the specific solution to this differential equation?
4. How would the solution change if an additional term, like $y = xy$, were added?
5. What are common methods to determine convergence intervals for solutions derived from power series?

Tip: In power series solutions, organizing terms carefully and matching indices is crucial for simplifying recurrence relations correctly.