solve : y" – 3y" + 2y = e 2x 1+e*
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 Statement:**
Solve the differential equation:
\[ y''' - 3y'' + 2y = \frac{e^{2x}}{1 + e^x} \]
**Explanation:**
This is a third-order linear non-homogeneous differential equation.
- The left-hand side consists of derivatives of \(y\):
- \( y''' \) denotes the third derivative of \( y \).
- \( -3y'' \) is three times the second derivative of \( y \).
- \( +2y \) is two times the original function \( y \).
- The right-hand side is a function \( \frac{e^{2x}}{1 + e^x} \), where:
- \( e^{2x} \) is the exponential function with base \( e \) raised to the power of \( 2x \).
- The denominator \( 1 + e^x \) is a polynomial shifted by an exponential function.
**Solution Approach:**
To solve this differential equation, consider the following steps:
1. **Find the Complementary Solution (Homogeneous Solution):**
- Solve \( y''' - 3y'' + 2y = 0 \) to find the complementary solution, \( y_c \).
2. **Find the Particular Solution:**
- Use an appropriate method (such as undetermined coefficients or variation of parameters) to find the particular solution to the non-homogeneous equation.
3. **General Solution:**
- Combine the complementary solution and the particular solution to form the general solution.
**Tools and Techniques:**
- The roots of the characteristic equation for the homogeneous part can help determine the form of the complementary solution.
- Integration techniques or the method of undetermined coefficients may be used to find a particular solution.
- Consider any initial conditions if provided for a complete solution.
This differential equation can generally be solved using software tools capable of symbolic mathematics if done analytically is complex.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F410816ed-4be2-4291-b0fb-4c8f4b2206ab%2Fbb018506-b12e-400c-8803-cb964c107cd6%2F6gpe0a_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Solve the differential equation:
\[ y''' - 3y'' + 2y = \frac{e^{2x}}{1 + e^x} \]
**Explanation:**
This is a third-order linear non-homogeneous differential equation.
- The left-hand side consists of derivatives of \(y\):
- \( y''' \) denotes the third derivative of \( y \).
- \( -3y'' \) is three times the second derivative of \( y \).
- \( +2y \) is two times the original function \( y \).
- The right-hand side is a function \( \frac{e^{2x}}{1 + e^x} \), where:
- \( e^{2x} \) is the exponential function with base \( e \) raised to the power of \( 2x \).
- The denominator \( 1 + e^x \) is a polynomial shifted by an exponential function.
**Solution Approach:**
To solve this differential equation, consider the following steps:
1. **Find the Complementary Solution (Homogeneous Solution):**
- Solve \( y''' - 3y'' + 2y = 0 \) to find the complementary solution, \( y_c \).
2. **Find the Particular Solution:**
- Use an appropriate method (such as undetermined coefficients or variation of parameters) to find the particular solution to the non-homogeneous equation.
3. **General Solution:**
- Combine the complementary solution and the particular solution to form the general solution.
**Tools and Techniques:**
- The roots of the characteristic equation for the homogeneous part can help determine the form of the complementary solution.
- Integration techniques or the method of undetermined coefficients may be used to find a particular solution.
- Consider any initial conditions if provided for a complete solution.
This differential equation can generally be solved using software tools capable of symbolic mathematics if done analytically is complex.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 7 steps with 7 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,
