y' = xy+x-y-1, y(0)=0
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
pleaxe show all srteps answer step by step
![**Differential Equations: Initial Value Problem**
In this exercise, we will solve an initial value problem involving the following first-order differential equation:
\[ y' = xy + x - y - 1, \quad y(0) = 0 \]
where:
- \( y' \) represents the derivative of \( y \) with respect to \( x \),
- \( x \) is the independent variable,
- \( y \) is the dependent variable.
### Steps to Solve the Initial Value Problem
1. **Identify the Type of Differential Equation**: Recognize that this is a first-order linear differential equation with an initial condition.
2. **Rewrite the Differential Equation**: We might be able to simplify or rewrite the equation for easier solving.
3. **Solve the Homogeneous Part**: Find the solution to the associated homogeneous equation.
4. **Find a Particular Solution**. Use methods such as undetermined coefficients or variation of parameters to find a particular solution to the non-homogeneous equation.
5. **Apply the Initial Condition**: Use \( y(0) = 0 \) to determine any constants of integration.
### Graphical Explanation
The equation and initial condition describe how the function \( y(x) \) evolves from the point \( (0,0) \).
By solving this differential equation, we can understand the behavior of \( y \) in relation to \( x \). This solution is essential in various fields such as physics, engineering, and other applied sciences where differential equations model real-world phenomena.
### Visualization
- **Graph of \( y(x) \)**: Plotting \( y(x) \) against \( x \) once the solution is obtained.
- **Direction Field**: This provides a graphical representation of the differential equation without finding its explicit solution by plotting slopes at grid points.
By understanding and solving this initial value problem, we gain insights into the dynamic system described by the differential equation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6aad8a00-c703-485c-bf76-99694ae31b4b%2F7e3e0b38-61d7-4e00-8213-7ad2c0875a0f%2Fkcf3uir_processed.png&w=3840&q=75)
Transcribed Image Text:**Differential Equations: Initial Value Problem**
In this exercise, we will solve an initial value problem involving the following first-order differential equation:
\[ y' = xy + x - y - 1, \quad y(0) = 0 \]
where:
- \( y' \) represents the derivative of \( y \) with respect to \( x \),
- \( x \) is the independent variable,
- \( y \) is the dependent variable.
### Steps to Solve the Initial Value Problem
1. **Identify the Type of Differential Equation**: Recognize that this is a first-order linear differential equation with an initial condition.
2. **Rewrite the Differential Equation**: We might be able to simplify or rewrite the equation for easier solving.
3. **Solve the Homogeneous Part**: Find the solution to the associated homogeneous equation.
4. **Find a Particular Solution**. Use methods such as undetermined coefficients or variation of parameters to find a particular solution to the non-homogeneous equation.
5. **Apply the Initial Condition**: Use \( y(0) = 0 \) to determine any constants of integration.
### Graphical Explanation
The equation and initial condition describe how the function \( y(x) \) evolves from the point \( (0,0) \).
By solving this differential equation, we can understand the behavior of \( y \) in relation to \( x \). This solution is essential in various fields such as physics, engineering, and other applied sciences where differential equations model real-world phenomena.
### Visualization
- **Graph of \( y(x) \)**: Plotting \( y(x) \) against \( x \) once the solution is obtained.
- **Direction Field**: This provides a graphical representation of the differential equation without finding its explicit solution by plotting slopes at grid points.
By understanding and solving this initial value problem, we gain insights into the dynamic system described by the differential equation.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 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,
