Find two power series solutions of the given differential equation about the ordinary point x = 0. y" + 8xy' + 8y = 0 Oy₁ = 1 - 4x²8x4 32 6 3 Oy₁ = 1 - 4x² + 8x4 - 32x6. 32 6 Oy₁ = 1 + 4x² - 8x4 + -x +... and y₂ = x - 3 Oy₁ = 1 - 4x² - 8x4 - -32x6 and Y₂ = X- Oy₁ = 1 - 4x² + 8x4 - 32 x6 - 3 + ... and y₂ = x- and y₂ = x + and y₂ = x + 83 64 5 + -X 15 00 m 8 -X + 3 8 3 8 8.3 3 64 5 + 15 + 64.5 -X + 15 64 5 -X 15 64 5 -X 15 + + 512 7 105 512 7 105 512 7 105 512 -X 105 512 7 105 + +... +... + -X + ***
Find two power series solutions of the given differential equation about the ordinary point x = 0. y" + 8xy' + 8y = 0 Oy₁ = 1 - 4x²8x4 32 6 3 Oy₁ = 1 - 4x² + 8x4 - 32x6. 32 6 Oy₁ = 1 + 4x² - 8x4 + -x +... and y₂ = x - 3 Oy₁ = 1 - 4x² - 8x4 - -32x6 and Y₂ = X- Oy₁ = 1 - 4x² + 8x4 - 32 x6 - 3 + ... and y₂ = x- and y₂ = x + and y₂ = x + 83 64 5 + -X 15 00 m 8 -X + 3 8 3 8 8.3 3 64 5 + 15 + 64.5 -X + 15 64 5 -X 15 64 5 -X 15 + + 512 7 105 512 7 105 512 7 105 512 -X 105 512 7 105 + +... +... + -X + ***
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Power Series Solutions for Differential Equations**
**Problem Statement:**
Find two power series solutions of the given differential equation about the ordinary point \( x = 0 \).
\[ y'' + 8xy' + 8y = 0 \]
**Solution Options:**
1. \[
\mathbf{y_1 = 1 - 4x^2 + 8x^4 - \frac{32}{3}x^6 - \cdots} \quad \text{and} \quad
\mathbf{y_2 = x - \frac{8}{3}x^3 + \frac{64}{15}x^5 - \frac{512}{105}x^7 + \cdots}
\]
2. \[
\mathbf{y_1 = 1 + 4x^2 + 8x^4 + \frac{32}{3}x^6 + \cdots} \quad \text{and} \quad
\mathbf{y_2 = x - \frac{8}{3}x^3 + \frac{64}{15}x^5 - \frac{512}{105}x^7 + \cdots}
\]
3. \[
\mathbf{y_1 = 1 - 4x^2 + 8x^4 - \frac{32}{3}x^6 - \cdots} \quad \text{and} \quad
\mathbf{y_2 = x + \frac{8}{3}x^3 + \frac{64}{15}x^5 + \frac{512}{105}x^7 + \cdots}
\]
4. \[
\mathbf{y_1 = 1 - 4x^2 + 8x^4 - \frac{32}{3}x^6 - \cdots} \quad \text{and} \quad
\mathbf{y_2 = x - \frac{8}{3}x^3 + \frac{64}{15}x^5 - \frac{512}{105}x^7 + \cdots}
\]
5. \[
\mathbf{y_1 = 1 - 4x^2 + 8x^4 - \frac{32}{3}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F995b5b40-26b1-420f-ae4c-d7493773d7f9%2F0349a51f-6940-4420-9044-2f5614d01197%2Foy2mli_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
**Power Series Solutions for Differential Equations**
**Problem Statement:**
Find two power series solutions of the given differential equation about the ordinary point \( x = 0 \).
\[ y'' + 8xy' + 8y = 0 \]
**Solution Options:**
1. \[
\mathbf{y_1 = 1 - 4x^2 + 8x^4 - \frac{32}{3}x^6 - \cdots} \quad \text{and} \quad
\mathbf{y_2 = x - \frac{8}{3}x^3 + \frac{64}{15}x^5 - \frac{512}{105}x^7 + \cdots}
\]
2. \[
\mathbf{y_1 = 1 + 4x^2 + 8x^4 + \frac{32}{3}x^6 + \cdots} \quad \text{and} \quad
\mathbf{y_2 = x - \frac{8}{3}x^3 + \frac{64}{15}x^5 - \frac{512}{105}x^7 + \cdots}
\]
3. \[
\mathbf{y_1 = 1 - 4x^2 + 8x^4 - \frac{32}{3}x^6 - \cdots} \quad \text{and} \quad
\mathbf{y_2 = x + \frac{8}{3}x^3 + \frac{64}{15}x^5 + \frac{512}{105}x^7 + \cdots}
\]
4. \[
\mathbf{y_1 = 1 - 4x^2 + 8x^4 - \frac{32}{3}x^6 - \cdots} \quad \text{and} \quad
\mathbf{y_2 = x - \frac{8}{3}x^3 + \frac{64}{15}x^5 - \frac{512}{105}x^7 + \cdots}
\]
5. \[
\mathbf{y_1 = 1 - 4x^2 + 8x^4 - \frac{32}{3}
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