2. 3. メーX, y0-2 | dx
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
solve the given initial-value problem
![### Differential Equation Solution Example
#### Problem Statement:
Solve the following differential equation with the given initial value:
\[
x y^2 \frac{dy}{dx} = y^3 - x^3, \quad y(1) = 2
\]
### Explanation:
1. **Understanding the equation:**
- The equation is given in the form \( x y^2 \frac{dy}{dx} = y^3 - x^3 \).
- This is a first-order, non-linear differential equation.
2. **Initial Condition:**
- The initial condition provided is \( y(1) = 2 \).
3. **Procedure:**
- To solve this, one might separate variables or use substitution methods, depending on what's most appropriate for the form of the differential equation.
### Detailed Steps:
1. Rewrite the differential equation:
\[
x y^2 \frac{dy}{dx} = y^3 - x^3
\]
Dividing both sides by \( y^2 \), we get:
\[
x \frac{dy}{dx} = y - \frac{x^3}{y^2}
\]
2. Multiplying both sides by \( dx \) and separating variables, we obtain:
\[
\frac{y^2}{y^3 - x^3} dy = \frac{1}{x} dx
\]
3. Integrate both sides within suitable limits.
### Conclusion:
- After integration, apply the initial condition \( y(1) = 2 \) to find the constant of integration.
- Solve the resulting equation to get the explicit form of \( y \) in terms of \( x \).
Please check an appropriate reference or differential equations resource for the integration steps and final solution. This equation might require more advanced integration techniques or numerical methods for a full solution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1e4fcbe0-ed74-4447-b38a-a2af13f9fabc%2F7e7d44df-8ef7-442b-9ba3-e0f9d566deb9%2Fb9fgp93_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Differential Equation Solution Example
#### Problem Statement:
Solve the following differential equation with the given initial value:
\[
x y^2 \frac{dy}{dx} = y^3 - x^3, \quad y(1) = 2
\]
### Explanation:
1. **Understanding the equation:**
- The equation is given in the form \( x y^2 \frac{dy}{dx} = y^3 - x^3 \).
- This is a first-order, non-linear differential equation.
2. **Initial Condition:**
- The initial condition provided is \( y(1) = 2 \).
3. **Procedure:**
- To solve this, one might separate variables or use substitution methods, depending on what's most appropriate for the form of the differential equation.
### Detailed Steps:
1. Rewrite the differential equation:
\[
x y^2 \frac{dy}{dx} = y^3 - x^3
\]
Dividing both sides by \( y^2 \), we get:
\[
x \frac{dy}{dx} = y - \frac{x^3}{y^2}
\]
2. Multiplying both sides by \( dx \) and separating variables, we obtain:
\[
\frac{y^2}{y^3 - x^3} dy = \frac{1}{x} dx
\]
3. Integrate both sides within suitable limits.
### Conclusion:
- After integration, apply the initial condition \( y(1) = 2 \) to find the constant of integration.
- Solve the resulting equation to get the explicit form of \( y \) in terms of \( x \).
Please check an appropriate reference or differential equations resource for the integration steps and final solution. This equation might require more advanced integration techniques or numerical methods for a full solution.
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 3 steps with 3 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.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,
