dy dt = (y² - 16) (1 - y)² = 2. (a) Find the long term solution for initial value y(0) (b) If y represents the temperature of an industrial boiler and t represents time, what is the range of initial values that may lead into catastrophic conditions (∞ or -∞ ).
dy dt = (y² - 16) (1 - y)² = 2. (a) Find the long term solution for initial value y(0) (b) If y represents the temperature of an industrial boiler and t represents time, what is the range of initial values that may lead into catastrophic conditions (∞ or -∞ ).
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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![**Differential Equations in Industrial Applications**
Consider the differential equation:
\[ \frac{dy}{dt} = (y^2 - 16)(1 - y)^2 \]
**Problem Statement:**
**(a)** Find the long-term solution for the initial value \( y(0) = 2 \).
**(b)** If \( y \) represents the temperature of an industrial boiler and \( t \) represents time, what is the range of initial values that may lead to catastrophic conditions (i.e., \( \pm \infty \))?
**Analysis:**
- **Part (a):** To solve for the long-term behavior of the solution when the initial value \( y(0) = 2 \), we need to analyze how the differential equation behaves over time.
- **Part (b):** Understanding the initial value conditions that could lead the temperature \( y \) to reach runaway values, either to \( \infty \) or \( -\infty \), helps in preventing catastrophic scenarios in industrial settings.
By analyzing the differential equation and its critical points, we can gain insights on the stability of the system and identify safe operating conditions for industrial processes.
**Graphical Insights:**
A detailed phase portrait or analytical examination could reveal the behavior of the solution \( y(t) \) over time and identify stable and unstable equilibria points. Safe initial conditions can then be determined by evaluating how perturbations from equilibrium evolve in the system.
Ensure to refer to methodological explanations for solving differential equations and determining stability in your educational resources.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F05981ae5-77e6-4403-91a6-173ce3b3a769%2F6dbc4650-3a1d-4386-bf9a-e7ad79e0d32c%2Fzwhi8pr_processed.png&w=3840&q=75)
Transcribed Image Text:**Differential Equations in Industrial Applications**
Consider the differential equation:
\[ \frac{dy}{dt} = (y^2 - 16)(1 - y)^2 \]
**Problem Statement:**
**(a)** Find the long-term solution for the initial value \( y(0) = 2 \).
**(b)** If \( y \) represents the temperature of an industrial boiler and \( t \) represents time, what is the range of initial values that may lead to catastrophic conditions (i.e., \( \pm \infty \))?
**Analysis:**
- **Part (a):** To solve for the long-term behavior of the solution when the initial value \( y(0) = 2 \), we need to analyze how the differential equation behaves over time.
- **Part (b):** Understanding the initial value conditions that could lead the temperature \( y \) to reach runaway values, either to \( \infty \) or \( -\infty \), helps in preventing catastrophic scenarios in industrial settings.
By analyzing the differential equation and its critical points, we can gain insights on the stability of the system and identify safe operating conditions for industrial processes.
**Graphical Insights:**
A detailed phase portrait or analytical examination could reveal the behavior of the solution \( y(t) \) over time and identify stable and unstable equilibria points. Safe initial conditions can then be determined by evaluating how perturbations from equilibrium evolve in the system.
Ensure to refer to methodological explanations for solving differential equations and determining stability in your educational resources.
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