Consider the differential equation x²y" - 7xy' +12y = 0; x², x6, (0, ∞0). Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. The functions satisfy the differential equation and are linearly independent since W(x², x6) = Form the general solution. y = #0 for 0 < x < 00.
Consider the differential equation x²y" - 7xy' +12y = 0; x², x6, (0, ∞0). Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. The functions satisfy the differential equation and are linearly independent since W(x², x6) = Form the general solution. y = #0 for 0 < x < 00.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Verify the Fundamental Set of Solutions
#### Differential Equation
Consider the differential equation:
\[ x^2 y'' - 7xy' + 12y = 0; \quad x^2, x^6, \quad (0, \infty). \]
Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval.
#### Linearly Independent Solutions
The functions satisfy the differential equation and are linearly independent since:
\[ W(x^2, x^6) = \]
\[ \neq 0 \quad \text{for} \quad 0 < x < \infty. \]
#### General Solution
Form the general solution:
\[ y = \]
\( y = C_1 x^2 + C_2 x^6 \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa4d7cfc5-c108-459e-b939-f2d2e6c37b5b%2F9cb47824-ce91-4bae-807d-856f4466edea%2F9axc70m_processed.png&w=3840&q=75)
Transcribed Image Text:### Verify the Fundamental Set of Solutions
#### Differential Equation
Consider the differential equation:
\[ x^2 y'' - 7xy' + 12y = 0; \quad x^2, x^6, \quad (0, \infty). \]
Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval.
#### Linearly Independent Solutions
The functions satisfy the differential equation and are linearly independent since:
\[ W(x^2, x^6) = \]
\[ \neq 0 \quad \text{for} \quad 0 < x < \infty. \]
#### General Solution
Form the general solution:
\[ y = \]
\( y = C_1 x^2 + C_2 x^6 \)
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