8. Solve the Cauchy-Euler equation 4x²y" + 8xy' + y = 0.
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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![### Problem 8: Cauchy-Euler Equation
**Objective:** Solve the Cauchy-Euler equation
\[ 4x^2y'' + 8xy' + y = 0. \]
### Explanation
The Cauchy-Euler equation is a type of linear differential equation of the form:
\[ a_n x^n y^{(n)} + a_{n-1} x^{n-1} y^{(n-1)} + \cdots + a_1 x y' + a_0 y = 0. \]
This particular problem provides a second-order Cauchy-Euler equation:
- **\( x^2 \) Coefficient: \( 4x^2 y'' \)**
- The second derivative \( y'' \) is multiplied by \( 4x^2 \).
- **\( x \) Coefficient: \( 8x y' \)**
- The first derivative \( y' \) is multiplied by \( 8x \).
- **Constant Term: \( y \)**
- The function \( y \) itself is included as is.
### Method of Solution
1. **Substitution:** Assume a solution of the form \( y = x^m \).
2. **Derivatives:** Compute the first and second derivatives:
- \( y' = mx^{m-1} \)
- \( y'' = m(m-1)x^{m-2} \)
3. **Substitute into the Original Equation:**
Replace \( y \), \( y' \), and \( y'' \) in the equation with the expressions derived:
\[
4x^2(m(m-1)x^{m-2}) + 8x(mx^{m-1}) + x^m = 0.
\]
4. **Simplify and Solve for \( m \):**
- Simplifies to:
\[
4m(m-1)x^m + 8mx^m + x^m = 0.
\]
- Factor out \( x^m \):
\[
x^m(4m(m-1) + 8m + 1) = 0.
\]
- Solve the characteristic equation:
\[
4m^2 + 4m + 1 = 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F007342aa-8bf1-438a-b2a8-d9b3fd40ac61%2F8df7518e-cefd-454e-9662-a68d708bc26c%2F2qr9iy_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 8: Cauchy-Euler Equation
**Objective:** Solve the Cauchy-Euler equation
\[ 4x^2y'' + 8xy' + y = 0. \]
### Explanation
The Cauchy-Euler equation is a type of linear differential equation of the form:
\[ a_n x^n y^{(n)} + a_{n-1} x^{n-1} y^{(n-1)} + \cdots + a_1 x y' + a_0 y = 0. \]
This particular problem provides a second-order Cauchy-Euler equation:
- **\( x^2 \) Coefficient: \( 4x^2 y'' \)**
- The second derivative \( y'' \) is multiplied by \( 4x^2 \).
- **\( x \) Coefficient: \( 8x y' \)**
- The first derivative \( y' \) is multiplied by \( 8x \).
- **Constant Term: \( y \)**
- The function \( y \) itself is included as is.
### Method of Solution
1. **Substitution:** Assume a solution of the form \( y = x^m \).
2. **Derivatives:** Compute the first and second derivatives:
- \( y' = mx^{m-1} \)
- \( y'' = m(m-1)x^{m-2} \)
3. **Substitute into the Original Equation:**
Replace \( y \), \( y' \), and \( y'' \) in the equation with the expressions derived:
\[
4x^2(m(m-1)x^{m-2}) + 8x(mx^{m-1}) + x^m = 0.
\]
4. **Simplify and Solve for \( m \):**
- Simplifies to:
\[
4m(m-1)x^m + 8mx^m + x^m = 0.
\]
- Factor out \( x^m \):
\[
x^m(4m(m-1) + 8m + 1) = 0.
\]
- Solve the characteristic equation:
\[
4m^2 + 4m + 1 = 0.
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