the following homogeneous DE. Find a fundamental set of solutions for the DE (1). Form the general solution of (1).
the following homogeneous DE. Find a fundamental set of solutions for the DE (1). Form the general solution of (1).
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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Consider the following homogeneous DE. Find a fundamental set of solutions for the DE (1). Form the general solution of (1).
![### Problem Statement:
Consider the differential equation:
\[ y'' + y' - 2y = 0. \quad \text{(1)} \]
- Find a fundamental set of solutions for the DE (1). Form the general solution of (1).
### Explanation:
In this problem, we are given a second-order linear homogeneous differential equation. The tasks are to:
1. Find a fundamental set of solutions.
2. Form the general solution based on the fundamental set of solutions.
Here's how you can approach this:
1. **Solve the characteristic equation**:
- The characteristic equation for a second-order DE in the form \( ay'' + by' + cy = 0 \) is \( ar^2 + br + c = 0 \).
- For the given DE (1): \( y'' + y' - 2y = 0 \), the characteristic equation is \( r^2 + r - 2 = 0 \).
2. **Find the roots of the characteristic equation**:
- Factoring the quadratic equation \( r^2 + r - 2 = 0 \): \((r - 1)(r + 2) = 0\).
- The roots are \( r_1 = 1 \) and \( r_2 = -2 \).
3. **Form the fundamental set of solutions**:
- The solutions corresponding to the roots \( r_1 \) and \( r_2 \) are \( y_1 = e^{r_1 x} \) and \( y_2 = e^{r_2 x} \).
- Therefore, the fundamental set of solutions is \( \{e^x, e^{-2x}\} \).
4. **Form the general solution**:
- The general solution of the DE is a linear combination of the fundamental set of solutions:
- \( y = C_1 e^x + C_2 e^{-2x} \), where \( C_1 \) and \( C_2 \) are arbitrary constants.
This process leads us to the general solution of the given differential equation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa061449e-1e75-4fa5-9e02-82a00bd019ab%2Fdabd06b5-8f9f-41c9-9843-f52864266f5f%2F5zpb5lo.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
Consider the differential equation:
\[ y'' + y' - 2y = 0. \quad \text{(1)} \]
- Find a fundamental set of solutions for the DE (1). Form the general solution of (1).
### Explanation:
In this problem, we are given a second-order linear homogeneous differential equation. The tasks are to:
1. Find a fundamental set of solutions.
2. Form the general solution based on the fundamental set of solutions.
Here's how you can approach this:
1. **Solve the characteristic equation**:
- The characteristic equation for a second-order DE in the form \( ay'' + by' + cy = 0 \) is \( ar^2 + br + c = 0 \).
- For the given DE (1): \( y'' + y' - 2y = 0 \), the characteristic equation is \( r^2 + r - 2 = 0 \).
2. **Find the roots of the characteristic equation**:
- Factoring the quadratic equation \( r^2 + r - 2 = 0 \): \((r - 1)(r + 2) = 0\).
- The roots are \( r_1 = 1 \) and \( r_2 = -2 \).
3. **Form the fundamental set of solutions**:
- The solutions corresponding to the roots \( r_1 \) and \( r_2 \) are \( y_1 = e^{r_1 x} \) and \( y_2 = e^{r_2 x} \).
- Therefore, the fundamental set of solutions is \( \{e^x, e^{-2x}\} \).
4. **Form the general solution**:
- The general solution of the DE is a linear combination of the fundamental set of solutions:
- \( y = C_1 e^x + C_2 e^{-2x} \), where \( C_1 \) and \( C_2 \) are arbitrary constants.
This process leads us to the general solution of the given differential equation.
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