Apply the method of reduction of order (not itť's formula) to find the second solution if 6y"+ y'-y = 0 y, = e3
Apply the method of reduction of order (not itť's formula) to find the second solution if 6y"+ y'-y = 0 y, = e3
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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![**Problem Statement:**
Apply the method of reduction of order (not its formula) to find the second solution for the differential equation:
\[ 6y'' + y' - y = 0 \]
given that one solution is
\[ y_1 = e^{\frac{x}{3}} \]
**Explanation:**
The task is to find a second linearly independent solution to the given second-order homogeneous linear differential equation using the method of reduction of order. This method involves assuming a solution of a specific form and using known solutions to reduce the order of the differential equation, making it easier to solve.
**Steps to Solve:**
1. Start with the known solution \( y_1 = e^{\frac{x}{3}} \).
2. To find a second solution \( y_2 \), assume \( y_2 = v(x) y_1 \), where \( v(x) \) is a function to be determined.
3. Substitute \( y_2 \) into the original differential equation to derive an equation for \( v(x) \).
4. Solve the resulting first-order differential equation to find the function \( v(x) \).
5. Multiply \( v(x) \) by \( y_1 \) to find the second solution \( y_2 \).
The process typically involves integration and algebraic manipulation, usually resulting in an expression that provides the general solution to the differential equation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4c1e8b90-227b-4a9c-b073-26e8337ecb0b%2F5d22396f-d82b-4e12-929d-24fafdbdad99%2F47asgdlr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Apply the method of reduction of order (not its formula) to find the second solution for the differential equation:
\[ 6y'' + y' - y = 0 \]
given that one solution is
\[ y_1 = e^{\frac{x}{3}} \]
**Explanation:**
The task is to find a second linearly independent solution to the given second-order homogeneous linear differential equation using the method of reduction of order. This method involves assuming a solution of a specific form and using known solutions to reduce the order of the differential equation, making it easier to solve.
**Steps to Solve:**
1. Start with the known solution \( y_1 = e^{\frac{x}{3}} \).
2. To find a second solution \( y_2 \), assume \( y_2 = v(x) y_1 \), where \( v(x) \) is a function to be determined.
3. Substitute \( y_2 \) into the original differential equation to derive an equation for \( v(x) \).
4. Solve the resulting first-order differential equation to find the function \( v(x) \).
5. Multiply \( v(x) \) by \( y_1 \) to find the second solution \( y_2 \).
The process typically involves integration and algebraic manipulation, usually resulting in an expression that provides the general solution to the differential equation.
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