How can i solve this exercise? **Power Series Expansion for Differential Equations**In this section, we will explore how to find at least the first four nonzero terms in a power series expansion about \(x_0\) for a general solution to a given differential equation, given the specified value for \(x_0\).Consider the differential equation given by:\[ y'' + (3x - 1) y' - y = 0 \]with the initial point \( x_0 = -1 \).To solve this differential equation using a power series expansion, we follow these steps:1. **Assume a Power Series Solution**: Assume that the solution \(y(x)\) can be expressed as a power series centered at \(x_0 = -1\): \[ y(x) = \sum_{n=0}^{\infty} a_n (x + 1)^n \]2. **Differentiating the Series**: To find \(y'(x)\) and \(y''(x)\), differentiate the power series term by term: \[ y'(x) = \sum_{n=1}^{\infty} a_n n (x + 1)^{n-1} \] \[ y''(x) = \sum_{n=2}^{\infty} a_n n (n-1) (x + 1)^{n-2} \]3. **Substituting into the Differential Equation**: Substitute \(y(x)\), \(y'(x)\), and \(y''(x)\) back into the differential equation: \[ \sum_{n=2}^{\infty} a_n n (n-1) (x + 1)^{n-2} + (3x - 1) \sum_{n=1}^{\infty} a_n n (x + 1)^{n-1} - \sum_{n=0}^{\infty} a_n (x + 1)^n = 0 \]4. **Combine Like Terms and Solve for Coefficients**: Align the powers of \((x + 1)\) and combine like terms. Setting coefficients of like powers of \((x + 1)\) to zero gives a recurrence relation for the coefficients \(a_n\).From this,

Name: How can i solve this exercise? **Power Series Expansion for Differenti
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Description: How can i solve this exercise? **Power Series Expansion for Differential Equations**In this section, we will explore how to find at least the first four nonzero terms in a power series expansion about \(x_0\) for a general solution to a given differe