Give an example of a second-order, constant-coefficient, homogeneous equation with non-real roots of the characteristic equation. Write down the linear system that corresponds to it. Can you find an example of an interconnected system of two tanks that is described by this system? Explain. Then solve the system.
Give an example of a second-order, constant-coefficient, homogeneous equation with non-real roots of the characteristic equation. Write down the linear system that corresponds to it. Can you find an example of an interconnected system of two tanks that is described by this system? Explain. Then solve the system.
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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![**Second-Order Homogeneous Equation Example**
**Problem Statement:**
1. **Equation Example:**
- Provide an example of a second-order, constant-coefficient, homogeneous equation with non-real roots of the characteristic equation.
2. **Corresponding Linear System:**
- Write down the linear system corresponding to this differential equation.
3. **Real-World Application:**
- Find an example of an interconnected system of two tanks that can be described by this system. Explain the relationship.
4. **Solution:**
- Solve the linear system described.
**Details and Explanation:**
- **Equation Example:**
An example of such an equation is:
\[
ay'' + by' + cy = 0
\]
Example with non-real roots:
\[
y'' + 2y' + 5y = 0
\]
- **Characteristic Equation:**
The characteristic equation is:
\[
r^2 + 2r + 5 = 0
\]
Solving gives non-real roots \( r = -1 \pm 2i \).
- **Corresponding Linear System:**
The equivalent system can be written as:
\[
\begin{align*}
x_1' &= x_2 \\
x_2' &= -5x_1 - 2x_2
\end{align*}
\]
This is derived by setting \( x_1 = y \) and \( x_2 = y' \).
- **Real-World Application:**
Consider two interconnected tanks where the rate of change of the liquid level in each tank can be modeled by the given system of equations. The liquid exchange between the tanks can mimic the behavior of the system described by the differential equation.
- **Solution of the System:**
To solve, use:
\[
x_1(t) = e^{-t} (C_1 \cos 2t + C_2 \sin 2t)
\]
\[
x_2(t) = e^{-t} ((-2C_1 + C_2)\cos 2t + (-C_1 - 2C_2)\sin 2t)
\]
where \( C_1 \) and \( C_2 \) are constants determined by initial](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe7e0ba88-0786-454c-b4b5-22240a9787f4%2F1dfbdc21-6eee-4881-82ef-16cb1531e859%2F8rpvk4n_processed.png&w=3840&q=75)
Transcribed Image Text:**Second-Order Homogeneous Equation Example**
**Problem Statement:**
1. **Equation Example:**
- Provide an example of a second-order, constant-coefficient, homogeneous equation with non-real roots of the characteristic equation.
2. **Corresponding Linear System:**
- Write down the linear system corresponding to this differential equation.
3. **Real-World Application:**
- Find an example of an interconnected system of two tanks that can be described by this system. Explain the relationship.
4. **Solution:**
- Solve the linear system described.
**Details and Explanation:**
- **Equation Example:**
An example of such an equation is:
\[
ay'' + by' + cy = 0
\]
Example with non-real roots:
\[
y'' + 2y' + 5y = 0
\]
- **Characteristic Equation:**
The characteristic equation is:
\[
r^2 + 2r + 5 = 0
\]
Solving gives non-real roots \( r = -1 \pm 2i \).
- **Corresponding Linear System:**
The equivalent system can be written as:
\[
\begin{align*}
x_1' &= x_2 \\
x_2' &= -5x_1 - 2x_2
\end{align*}
\]
This is derived by setting \( x_1 = y \) and \( x_2 = y' \).
- **Real-World Application:**
Consider two interconnected tanks where the rate of change of the liquid level in each tank can be modeled by the given system of equations. The liquid exchange between the tanks can mimic the behavior of the system described by the differential equation.
- **Solution of the System:**
To solve, use:
\[
x_1(t) = e^{-t} (C_1 \cos 2t + C_2 \sin 2t)
\]
\[
x_2(t) = e^{-t} ((-2C_1 + C_2)\cos 2t + (-C_1 - 2C_2)\sin 2t)
\]
where \( C_1 \) and \( C_2 \) are constants determined by initial
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