This is the third time I've asked this, the system of two tanks NEEDS to be described by a second-order differential eq
This is the third time I've asked this, the system of two tanks NEEDS to be described by a second-order differential eq
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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This is the third time I've asked this, the system of two tanks NEEDS to be described by a second-order
![**Example of a Second-Order Differential Equation and System of Tanks**
**Problem Statement:**
1. **Second-Order Differential Equation:**
- Give an example of a second-order, constant-coefficient, homogeneous equation with non-real roots of the characteristic equation.
- *Task:* Write down the linear system corresponding to it.
2. **Interconnected System of Tanks:**
- Find an example of an interconnected system of two tanks described by this system.
- *Task:* Explain and solve the system.
---
**Solution Approach:**
1. **Example of a Second-Order Differential Equation:**
Consider the differential equation:
\[
y'' + 2y' + 5y = 0
\]
- **Characteristic Equation:** By substituting \( y = e^{\lambda t} \), we derive the characteristic equation:
\[
\lambda^2 + 2\lambda + 5 = 0
\]
- **Roots:** Solving this quadratic equation gives non-real, complex roots:
\[
\lambda = -1 \pm 2i
\]
2. **Interconnected System of Tanks:**
- **Description:** Imagine two tanks connected in series with water flowing between them. The input to the first tank and the output from the second tank control the system's dynamics.
- **Modeling the System:** The rate of change of water in each tank can be described by a system of linear differential equations. Using the given conditions and structure, the system can be represented in a form similar to:
\[
\begin{align*}
\frac{dx_1}{dt} &= -a_{11}x_1 + a_{12}x_2, \\
\frac{dx_2}{dt} &= a_{21}x_1 - a_{22}x_2.
\end{align*}
\]
where constants \(a_{11}, a_{12}, a_{21},\) and \(a_{22}\) are defined based on the system's specifics.
- **Solve the System:** Use techniques such as matrix methods or Laplace transforms to solve the above system, obtaining expressions for \(x_1(t)\) and \(x_2(t)\).
---
This example illustrates the concept of second-order differential equations with complex roots in a](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe7e0ba88-0786-454c-b4b5-22240a9787f4%2F40fc47eb-c560-483a-bff9-72e6a6e03569%2Fw2622go_processed.png&w=3840&q=75)
Transcribed Image Text:**Example of a Second-Order Differential Equation and System of Tanks**
**Problem Statement:**
1. **Second-Order Differential Equation:**
- Give an example of a second-order, constant-coefficient, homogeneous equation with non-real roots of the characteristic equation.
- *Task:* Write down the linear system corresponding to it.
2. **Interconnected System of Tanks:**
- Find an example of an interconnected system of two tanks described by this system.
- *Task:* Explain and solve the system.
---
**Solution Approach:**
1. **Example of a Second-Order Differential Equation:**
Consider the differential equation:
\[
y'' + 2y' + 5y = 0
\]
- **Characteristic Equation:** By substituting \( y = e^{\lambda t} \), we derive the characteristic equation:
\[
\lambda^2 + 2\lambda + 5 = 0
\]
- **Roots:** Solving this quadratic equation gives non-real, complex roots:
\[
\lambda = -1 \pm 2i
\]
2. **Interconnected System of Tanks:**
- **Description:** Imagine two tanks connected in series with water flowing between them. The input to the first tank and the output from the second tank control the system's dynamics.
- **Modeling the System:** The rate of change of water in each tank can be described by a system of linear differential equations. Using the given conditions and structure, the system can be represented in a form similar to:
\[
\begin{align*}
\frac{dx_1}{dt} &= -a_{11}x_1 + a_{12}x_2, \\
\frac{dx_2}{dt} &= a_{21}x_1 - a_{22}x_2.
\end{align*}
\]
where constants \(a_{11}, a_{12}, a_{21},\) and \(a_{22}\) are defined based on the system's specifics.
- **Solve the System:** Use techniques such as matrix methods or Laplace transforms to solve the above system, obtaining expressions for \(x_1(t)\) and \(x_2(t)\).
---
This example illustrates the concept of second-order differential equations with complex roots in a
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