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Let yı and y2 be two solutions of A(x)y" + B(x)y' + C(x)y = 0 on an open interval I where A, B, and C are continuous and A(x) is never zero. (a) Let W = W(y1. y2). Show that A(x) dx = (y1)(4y%) – (y2)(4y“). Then substitute for Ay, and Ay{ from the original differ- ential equation to show that A(x). dx - B(x)W(x). (b) Solve this first-order equation to deduce Abel's for- mula W(x) = K exp (- / dx). A(x) where K is a constant. (c) Why does Abel's formula imply that the Wronskian W(y1. y2) is either zero every- where or nonzero everywhere (as stated in Theorem 3)?

Let yı and y2 be two solutions of A(x)y" + B(x)y' + C(x)y = 0 on an open interval I where A, B, and C are continuous and A(x) is never zero. (a) Let W = W(y1. y2). Show that A(x) dx = (y1)(4y%) – (y2)(4y“). Then substitute for Ay, and Ay{ from the original differ- ential equation to show that A(x). dx - B(x)W(x). (b) Solve this first-order equation to deduce Abel's for- mula W(x) = K exp (- / dx). A(x) where K is a constant. (c) Why does Abel's formula imply that the Wronskian W(y1. y2) is either zero every- where or nonzero everywhere (as stated in Theorem 3)?

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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Let yı and y2 be two solutions of A(x)y" + B(x)y' + C(x)y = 0 on an open interval I where A, B, and C are continuous and A(x) is never zero. (a) Let W = W(y1. y2). Show that A(x) dx = (y1)(4y%) – (y2)(4y“). Then substitute for Ay, and Ay{ from the original differ- ential equation to show that A(x). dx - B(x)W(x). (b) Solve this first-order equation to deduce Abel's for- mula W(x) = K exp (- / dx). A(x) where K is a constant. (c) Why does Abel's formula imply that the Wronskian W(y1. y2) is either zero every- where or nonzero everywhere (as stated in Theorem 3)?
Transcribed Image Text:Let yı and y2 be two solutions of A(x)y" + B(x)y' + C(x)y = 0 on an open interval I where A, B, and C are continuous and A(x) is never zero. (a) Let W = W(y1. y2). Show that A(x) dx = (y1)(4y%) – (y2)(4y“). Then substitute for Ay, and Ay{ from the original differ- ential equation to show that A(x). dx - B(x)W(x). (b) Solve this first-order equation to deduce Abel's for- mula W(x) = K exp (- / dx). A(x) where K is a constant. (c) Why does Abel's formula imply that the Wronskian W(y1. y2) is either zero every- where or nonzero everywhere (as stated in Theorem 3)?
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