Verify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval. 2x2y" + 5xy + y = x² = x; y = ₁x¹/² + ₂x¹+x²x₁ (0,00) The functions x-1/2 and x-1 satisfy the differential equation and are linearly independent since W(x-1/2, x-1)= Yp= is a particular solution of the nonhomogeneous equation. *0 for 0 < x <∞. So the functions x-1/2 and x-1 form a fundamental set of solutions of the associated homogeneous equation, and
Verify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval. 2x2y" + 5xy + y = x² = x; y = ₁x¹/² + ₂x¹+x²x₁ (0,00) The functions x-1/2 and x-1 satisfy the differential equation and are linearly independent since W(x-1/2, x-1)= Yp= is a particular solution of the nonhomogeneous equation. *0 for 0 < x <∞. So the functions x-1/2 and x-1 form a fundamental set of solutions of the associated homogeneous equation, and
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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**General Solution of Nonhomogeneous Differential Equations**
**Problem Statement:**
Verify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval.
\[ 2x^2y'' + 5xy' + y = x^2 - x; \; y = c_1 x^{-1/2} + c_2 x^{-1} + \frac{1}{15}x^2 - \frac{1}{6}x \; (0, \infty) \]
**Solution Explanation:**
1. The functions \( x^{-1/2} \) and \( x^{-1} \) satisfy the differential equation and are linearly independent since:
\[ W(x^{-1/2}, x^{-1}) = \]
\[ \boxed{} \]
is not equal to 0 for \( 0 < x < \infty \). Therefore, the functions \( x^{-1/2} \) and \( x^{-1} \) form a fundamental set of solutions of the associated homogeneous equation.
2. The particular solution \( y_P \) of the nonhomogeneous equation is given by:
\[ y_P = \boxed{} \]
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Transcribed Image Text:---
**General Solution of Nonhomogeneous Differential Equations**
**Problem Statement:**
Verify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval.
\[ 2x^2y'' + 5xy' + y = x^2 - x; \; y = c_1 x^{-1/2} + c_2 x^{-1} + \frac{1}{15}x^2 - \frac{1}{6}x \; (0, \infty) \]
**Solution Explanation:**
1. The functions \( x^{-1/2} \) and \( x^{-1} \) satisfy the differential equation and are linearly independent since:
\[ W(x^{-1/2}, x^{-1}) = \]
\[ \boxed{} \]
is not equal to 0 for \( 0 < x < \infty \). Therefore, the functions \( x^{-1/2} \) and \( x^{-1} \) form a fundamental set of solutions of the associated homogeneous equation.
2. The particular solution \( y_P \) of the nonhomogeneous equation is given by:
\[ y_P = \boxed{} \]
---
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