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46. Assume that all functions in this exercise are defined on a common interval (a, b). (a) Prove: If y₁ and y2 are solutions of and y' + p(x)y = f(x) y' + p(x)y=f2(x) respectively, and c₁ and c₂ are constants, then y = c191 + €22 is a solution of y+p(x)y=c1f1(x) + €2 ƒ2(x). (This is the principle of superposition.) (b) Use (a) to show that if y₁ and y2 are solutions of the nonhomogeneous equation then yı - 2 is a solution of the homogeneous equation y+p(x)y = f(x), (A) y+p(x)y=0. (B) (c) Use (a) to show that if y₁ is a solution of (A) and y2 is a solution of (B), then y₁ + y2 is a solution of (A).

46. Assume that all functions in this exercise are defined on a common interval (a, b). (a) Prove: If y₁ and y2 are solutions of and y' + p(x)y = f(x) y' + p(x)y=f2(x) respectively, and c₁ and c₂ are constants, then y = c191 + €22 is a solution of y+p(x)y=c1f1(x) + €2 ƒ2(x). (This is the principle of superposition.) (b) Use (a) to show that if y₁ and y2 are solutions of the nonhomogeneous equation then yı - 2 is a solution of the homogeneous equation y+p(x)y = f(x), (A) y+p(x)y=0. (B) (c) Use (a) to show that if y₁ is a solution of (A) and y2 is a solution of (B), then y₁ + y2 is a solution of (A).

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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46. Assume that all functions in this exercise are defined on a common interval (a, b). (a) Prove: If y₁ and y2 are solutions of and y' + p(x)y = f(x) y' + p(x)y=f2(x) respectively, and c₁ and c₂ are constants, then y = c191 + €22 is a solution of y+p(x)y=c1f1(x) + €2 ƒ2(x). (This is the principle of superposition.) (b) Use (a) to show that if y₁ and y2 are solutions of the nonhomogeneous equation then yı - 2 is a solution of the homogeneous equation y+p(x)y = f(x), (A) y+p(x)y=0. (B) (c) Use (a) to show that if y₁ is a solution of (A) and y2 is a solution of (B), then y₁ + y2 is a solution of (A).
Transcribed Image Text:46. Assume that all functions in this exercise are defined on a common interval (a, b). (a) Prove: If y₁ and y2 are solutions of and y' + p(x)y = f(x) y' + p(x)y=f2(x) respectively, and c₁ and c₂ are constants, then y = c191 + €22 is a solution of y+p(x)y=c1f1(x) + €2 ƒ2(x). (This is the principle of superposition.) (b) Use (a) to show that if y₁ and y2 are solutions of the nonhomogeneous equation then yı - 2 is a solution of the homogeneous equation y+p(x)y = f(x), (A) y+p(x)y=0. (B) (c) Use (a) to show that if y₁ is a solution of (A) and y2 is a solution of (B), then y₁ + y2 is a solution of (A).
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