(2x+x³)y"-y'-6xy=0 solve by power series solutions 

We are tasked with solving the differential equation:

\[
(2x + x^3)y'' - y' - 6xy = 0
\]

using a power series solution.

### Step 1: Assume a power series solution for \( y(x) \)
We assume the solution for \( y(x) \) in the form of a power series around \( x = 0 \):

\[
y(x) = \sum_{n=0}^{\infty} a_n x^n
\]

### Step 2: Derivatives of \( y(x) \)
To substitute into the given differential equation, we need the first and second derivatives of \( y(x) \).

\[
y'(x) = \sum_{n=1}^{\infty} n a_n x^{n-1}
\]
\[
y''(x) = \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2}
\]

### Step 3: Substitute into the differential equation
Substitute \( y(x) \), \( y'(x) \), and \( y''(x) \) into the given equation \( (2x + x^3)y'' - y' - 6xy = 0 \).

- The term \( (2x + x^3)y'' \) becomes:
  \[
  (2x + x^3) \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2} = 2x \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2} + x^3 \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2}
  \]
  
  Breaking these terms up:
  \[
  2x \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2} = \sum_{n=2}^{\infty} 2n(n-1) a_n x^{n-1}
  \]
  \[
  x^3 \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2} = \sum_{n=2}^{\infty} n(n-1) a_n x^{n+1}
  \]

- The term \( -y' \) becomes:
  \[
  - \sum_{n=1}^{\infty} n a_n x^{n-1}
  \]

- The term \( -6xy \) becomes:
  \[
  -6x \sum_{n=0}^{\infty} a_n x^n = - \sum_{n=0}^{\infty} 6 a_n x^{n+1}
  \]

### Step 4: Combine like powers of \( x \)
Now, we combine all the terms together and collect powers of \( x \).

For the \( x^0 \) term:
\[
0 \text{ (no contribution from the other terms)}
\]

For the \( x^1 \) term:
\[
2a_2 - a_1 = 0
\]
which gives \( a_2 = \frac{a_1}{2} \).

For the \( x^n \) term (\( n \geq 2 \)):
\[
2n(n-1) a_n + a_{n+2} - n a_n - 6a_{n-1} = 0
\]
This is the recurrence relation that will help us solve for the coefficients \( a_n \).

### Step 5: Solve the recurrence relation
The recurrence relation is:

\[
2n(n-1) a_n + a_{n+2} - n a_n - 6a_{n-1} = 0
\]

For specific values of \( n \), this relation will generate the coefficients \( a_n \).

Let me know if you want to expand on the steps to find the coefficients in more detail.

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(2x + x³)y'' - y' - 6xy = 0 solve by power series solutions

Explore how to solve the differential equation (2x + x³)y'' - y' - 6xy = 0 using power series solutions. This guide walks through the power series expansion, calculating derivatives, substituting them into the differential equation, and deriving the recurrence relation for coefficients. Suitable for undergraduate mathematics students.

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