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Step 1: Express the solution as a power series: $y(x) = \sum_{n=0}^\infty a_nx^{n+r}$. Step 2: Substitute into the differential equation and match powers of $x$. Step 3: Find the indicial equation and solve it for $r$. Step 4: Use the resulting recursion relation to find $a_n$. Series solution: $y(x) = x^r \sum_{n=0}^\infty a_nx^n$, with $a_n$ calculated from the recursion formula.

Step 1: Assume a power series solution $y(x) = \sum_{n=0}^\infty a_nx^{n+r}$. Step 2: Calculate derivatives and substitute into the given equation. Step 3: Obtain the indicial equation and solve for $r$. Step 4: Apply recurrence relation to determine the coefficients $a_n$. Series solution: $y(x) = x^{r}(a_0 + a_1x + a_2x^2 + ...)$ for the values of $r$ that satisfy the indicial equation.

Step 1: Assume a power series solution $y(x) = \sum_{n=0}^\infty a_nx^{n+r}$. Step 2: Calculate derivatives and substitute into the given equation. Step 3: Determine indicial equation by considering the lowest powers of x. Step 4: Find the coefficients $a_n$ using a recursion formula. Series solution: $y(x) = (a_0 + a_1x + ... )x^r$ where $r$ is a root of the indicial equation.

Step 1: Assume $y(x) = \sum_{n=0}^\infty a_nx^{n+r}$. Step 2: Derive $y'$ and $y''$ and plug into the equation. Step 3: Formulate indicial equation and find the values of $r$. Step 4: Obtain coefficients using the recurrence formula. Final series solution: $y(x) = (a_0 + a_1x + ... )x^r$ where $r$ is found from the indicial equation.

Step 1: Write the power series solution as $y(x) = \sum_{n=0}^\infty a_nx^{n+r}$. Step 2: Substitute the series into the equation and align the terms by powers of $x$. Step 3: Derive the indicial equation and determine root $r$. Step 4: Find the coefficients through the recursive relationship. Solution: $y(x) = x^{r}(a_0 + a_1x + a_2x^2 + ...)$, with $a_n$ found using the recursion relation.

Step 1: Formulate $y(x) = \sum_{n=0}^\infty a_nx^{n+r}$. Step 2: Differentiate and plug the series into the DE. Step 3: Indicial equation is found by setting the lowest power of $x$ to zero. Step 4: Use the recursive relationship for $a_n$ to find each term. Solution: Series $y(x) = a_0x^r + a_1x^{r+1} + ...$, with $a_n$ from recursion.

Step 1: Write $y(x) = \sum_{n=0}^\infty a_nx^{n+r}$. Step 2: Substitute the series for $y$, $y'$, and $y''$ into the equation. Step 3: Calculate the indicial equation and solve for $r$. Step 4: Use the recurrence relation for $a_n$ to define all coefficients. Series solution: $y(x) = x^r(a_0 + a_1x + a_2x^2 + ...)$, and coefficients are determined by the recursion.

Step 1: Suggest $y(x) = \sum_{n=0}^\infty a_nx^{n+r}$. Step 2: Insert the series representation of $y$ and its derivatives into the DE. Step 3: Use the fact that coefficients of each power of $x$ must independently vanish to find the indicial equation. Step 4: Solve for the coefficients using the resulting recursive relationship. Solution: $y(x) = x^{r}(a_0 + a_1x + a_2x^2 + ...)$ where the recursion determines $a_n$.

Step 1: Assume a series solution: $y(x) = \sum_{n=0}^\infty a_nx^{n+r}$. Step 2: Substitute $y$, $y'$, and $y''$ from the series into the DE. Step 3: Find the indicial equation and solve. Step 4: Calculate the coefficients using the recursion given by the DE. Series solution: $y(x) = x^{r}(a_0 + a_1x + a_2x^2 + ...)$ where the $a_n$'s follow the recursive formula.