6. For what values of y(0) will the solution of y' = y(y – 2),x > 0 a) be a constant function b) be an increasing function c) be an decreasing function (hint: look at the direction field)
6. For what values of y(0) will the solution of y' = y(y – 2),x > 0 a) be a constant function b) be an increasing function c) be an decreasing function (hint: look at the direction field)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Problem 6: Determining the Behavior of Solutions**
Consider the differential equation:
\[ y' = y(y - 2), \quad x \geq 0 \]
For what values of \( y(0) \) will the solution:
a) Be a constant function
b) Be an increasing function
c) Be a decreasing function
*(Hint: Look at the direction field.)*
**Explanation:**
This problem involves analyzing a first-order differential equation to determine the behavior of its solutions based on the initial value \( y(0) \).
- **Direction Field Insight:** The direction field provides a visual representation of the slope of solutions at various points. By analyzing this, we can infer whether solutions are constant, increasing, or decreasing for different initial values \( y(0) \).
To solve the problem:
- **Constant Function:** Check when the right-hand side of the equation \( y(y - 2) \) is zero.
- **Increasing Function:** Determine when the derivative \( y' \) is positive.
- **Decreasing Function:** Determine when the derivative \( y' \) is negative.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F91ff5dda-0dcd-4838-ba9e-df212171ecd4%2F195629c5-9674-41b0-a01f-f87fff0748a6%2Fw3sfzgi_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 6: Determining the Behavior of Solutions**
Consider the differential equation:
\[ y' = y(y - 2), \quad x \geq 0 \]
For what values of \( y(0) \) will the solution:
a) Be a constant function
b) Be an increasing function
c) Be a decreasing function
*(Hint: Look at the direction field.)*
**Explanation:**
This problem involves analyzing a first-order differential equation to determine the behavior of its solutions based on the initial value \( y(0) \).
- **Direction Field Insight:** The direction field provides a visual representation of the slope of solutions at various points. By analyzing this, we can infer whether solutions are constant, increasing, or decreasing for different initial values \( y(0) \).
To solve the problem:
- **Constant Function:** Check when the right-hand side of the equation \( y(y - 2) \) is zero.
- **Increasing Function:** Determine when the derivative \( y' \) is positive.
- **Decreasing Function:** Determine when the derivative \( y' \) is negative.
Expert Solution
Step 1
First of all we draw the direction field of the differential equation
y' = y(y-2)
Step by step
Solved in 2 steps with 1 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
