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Show that solutions to 2nd order linear equation y"+p(t)y'+q(t)y=0 form a vector space through showing closure under addition and scalar multiplication.

Show that solutions to 2nd order linear equation y"+p(t)y'+q(t)y=0 form a vector space through showing closure under addition and scalar multiplication.

Calculus: Early Transcendentals
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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**Understanding Vector Spaces in the Context of Second-Order Linear Differential Equations** To demonstrate that the solutions to the second-order linear differential equation \( y'' + p(t)y' + q(t)y = 0 \) form a vector space, we must show that the set of solutions satisfies two key properties: closure under addition and closure under scalar multiplication. ### Closure Under Addition This property requires that if \( y_1(t) \) and \( y_2(t) \) are both solutions of the differential equation, then their sum \( y_1(t) + y_2(t) \) must also be a solution. ### Closure Under Scalar Multiplication For closure under scalar multiplication, we must confirm that if \( y(t) \) is a solution and \( c \) is a scalar, then the function \( c \cdot y(t) \) also satisfies the differential equation. By verifying these properties, we establish that the solutions form a vector space, providing a robust framework for understanding the behavior and combination of solutions in this mathematical context.
Transcribed Image Text:**Understanding Vector Spaces in the Context of Second-Order Linear Differential Equations** To demonstrate that the solutions to the second-order linear differential equation \( y'' + p(t)y' + q(t)y = 0 \) form a vector space, we must show that the set of solutions satisfies two key properties: closure under addition and closure under scalar multiplication. ### Closure Under Addition This property requires that if \( y_1(t) \) and \( y_2(t) \) are both solutions of the differential equation, then their sum \( y_1(t) + y_2(t) \) must also be a solution. ### Closure Under Scalar Multiplication For closure under scalar multiplication, we must confirm that if \( y(t) \) is a solution and \( c \) is a scalar, then the function \( c \cdot y(t) \) also satisfies the differential equation. By verifying these properties, we establish that the solutions form a vector space, providing a robust framework for understanding the behavior and combination of solutions in this mathematical context.
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