If y = 2² is one solution of x²y" + 2xy' − 6y = 0, find the other linearly independent solution.
If y = 2² is one solution of x²y" + 2xy' − 6y = 0, find the other linearly independent solution.
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:**
If \( y = x^2 \) is one solution of
\[ x^2 y'' + 2xy' - 6y = 0, \]
find the other linearly independent solution.
---
**Detailed Explanation:**
In this problem, you are given a second-order linear differential equation with \( y = x^2 \) as one of its solutions. The general form of the equation is:
\[ x^2 y'' + 2xy' - 6y = 0. \]
To find the other linearly independent solution, we can use the method of reduction of order. This method involves finding another solution, \( y_2 \), that is linearly independent of the given solution \( y_1 = x^2 \).
To apply the reduction of order technique:
1. Assume that the second solution \( y_2 \) has the form:
\[ y_2 = v(x) y_1 = v(x) x^2, \]
where \( v(x) \) is an unknown function to be determined.
2. Calculate the first derivative \( y_2' \) using the product rule:
\[ y_2' = (vx^2)' = v' x^2 + v(2x) = v' x^2 + 2vx. \]
3. Calculate the second derivative \( y_2'' \):
\[ y_2'' = (v' x^2 + 2vx)' = (v' x^2)' + (2vx)' = v'' x^2 + 2v' x + 2v + 2v'. \]
4. Substitute \( y_2 \), \( y_2' \), and \( y_2'' \) into the original differential equation:
\[ x^2 (v'' x^2 + 2v' x + 2v + 2v') + 2x (v' x^2 + 2vx) - 6v x^2 = 0. \]
5. Simplify the resulting equation to solve for \( v(x) \).
Through careful computation and simplification, you will find that the second solution is generally of a different form that involves logarithmic or polynomial terms, depending on the structure of the homogeneous linear differential equation.
By finding \( v(x) \), you can determine the second linearly independent solution and complete](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8bdf6582-40e2-47b9-bd75-44351bba3f76%2F6bc22731-ad60-4780-b7df-3ee63cba298b%2F6h3hrl_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
If \( y = x^2 \) is one solution of
\[ x^2 y'' + 2xy' - 6y = 0, \]
find the other linearly independent solution.
---
**Detailed Explanation:**
In this problem, you are given a second-order linear differential equation with \( y = x^2 \) as one of its solutions. The general form of the equation is:
\[ x^2 y'' + 2xy' - 6y = 0. \]
To find the other linearly independent solution, we can use the method of reduction of order. This method involves finding another solution, \( y_2 \), that is linearly independent of the given solution \( y_1 = x^2 \).
To apply the reduction of order technique:
1. Assume that the second solution \( y_2 \) has the form:
\[ y_2 = v(x) y_1 = v(x) x^2, \]
where \( v(x) \) is an unknown function to be determined.
2. Calculate the first derivative \( y_2' \) using the product rule:
\[ y_2' = (vx^2)' = v' x^2 + v(2x) = v' x^2 + 2vx. \]
3. Calculate the second derivative \( y_2'' \):
\[ y_2'' = (v' x^2 + 2vx)' = (v' x^2)' + (2vx)' = v'' x^2 + 2v' x + 2v + 2v'. \]
4. Substitute \( y_2 \), \( y_2' \), and \( y_2'' \) into the original differential equation:
\[ x^2 (v'' x^2 + 2v' x + 2v + 2v') + 2x (v' x^2 + 2vx) - 6v x^2 = 0. \]
5. Simplify the resulting equation to solve for \( v(x) \).
Through careful computation and simplification, you will find that the second solution is generally of a different form that involves logarithmic or polynomial terms, depending on the structure of the homogeneous linear differential equation.
By finding \( v(x) \), you can determine the second linearly independent solution and complete
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