Find the Taylor polynomial of order 5 at 0 generated by the solution of the initial value y" + x?y – 4y = 0, y(0) = 2, y'(0) = 5. %3D

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Supposed to apply power series to solve where y(x) = sum from 0 to infinity of Cnxn

**Problem Statement:**

Find the Taylor polynomial of order 5 at 0 generated by the solution of the initial value problem:

\[ y'' + x^2 y' - 4y = 0, \quad y(0) = 2, \quad y'(0) = 5. \]

**Explanation:**

In this problem, we are asked to determine the Taylor polynomial of order 5 for the function \( y \), which is the solution to the given differential equation with the specified initial conditions. The Taylor polynomial will be centered at \( x = 0 \). 

Here's a step-by-step guide on how to solve it:

1. **Formulating the Taylor Series:**

   The Taylor series of a function \( y(x) \) around \( x = 0 \) is given by:
   
   \[
   T_5(x) = y(0) + y'(0)x + \frac{y''(0)}{2!}x^2 + \frac{y'''(0)}{3!}x^3 + \frac{y''''(0)}{4!}x^4 + \frac{y'''''(0)}{5!}x^5
   \]

2. **Initial Conditions:**

   From the given initial conditions, we know:
   
   \[
   y(0) = 2
   \]
   \[
   y'(0) = 5
   \]

3. **Calculating Higher-Order Derivatives:**

   To complete the Taylor polynomial up to order 5, we need to determine \( y''(0) \), \( y'''(0) \), \( y''''(0) \), and \( y'''''(0) \). This requires differentiating the given differential equation and evaluating at \( x = 0 \).

4. **Using the Differential Equation:**

   The differential equation is:
   
   \[
   y'' + x^2 y' - 4y = 0
   \]
   
   - At \( x = 0 \):
     \[
     y''(0) - 4y(0) = 0
     \]
     \[
     y''(0) = 4y(0) = 4 \cdot 2 = 8
     \]
   
   - Differentiate the equation:
     \
Transcribed Image Text:**Problem Statement:** Find the Taylor polynomial of order 5 at 0 generated by the solution of the initial value problem: \[ y'' + x^2 y' - 4y = 0, \quad y(0) = 2, \quad y'(0) = 5. \] **Explanation:** In this problem, we are asked to determine the Taylor polynomial of order 5 for the function \( y \), which is the solution to the given differential equation with the specified initial conditions. The Taylor polynomial will be centered at \( x = 0 \). Here's a step-by-step guide on how to solve it: 1. **Formulating the Taylor Series:** The Taylor series of a function \( y(x) \) around \( x = 0 \) is given by: \[ T_5(x) = y(0) + y'(0)x + \frac{y''(0)}{2!}x^2 + \frac{y'''(0)}{3!}x^3 + \frac{y''''(0)}{4!}x^4 + \frac{y'''''(0)}{5!}x^5 \] 2. **Initial Conditions:** From the given initial conditions, we know: \[ y(0) = 2 \] \[ y'(0) = 5 \] 3. **Calculating Higher-Order Derivatives:** To complete the Taylor polynomial up to order 5, we need to determine \( y''(0) \), \( y'''(0) \), \( y''''(0) \), and \( y'''''(0) \). This requires differentiating the given differential equation and evaluating at \( x = 0 \). 4. **Using the Differential Equation:** The differential equation is: \[ y'' + x^2 y' - 4y = 0 \] - At \( x = 0 \): \[ y''(0) - 4y(0) = 0 \] \[ y''(0) = 4y(0) = 4 \cdot 2 = 8 \] - Differentiate the equation: \
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