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Get Full Access to Fundamentals Of Differential Equations - 8 Edition - Chapter 4.7 - Problem 53e
Get Full Access to Fundamentals Of Differential Equations - 8 Edition - Chapter 4.7 - Problem 53e

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# os. Let (t) be a solution to y’’+py’+qy = 0 on (a, b),

ISBN: 9780321747730 43

## Solution for problem 53E Chapter 4.7

Fundamentals of Differential Equations | 8th Edition

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Problem 53E

Isolated Zeros. Let ?(t) be a solution to y’’+py’+qy = 0 on (a, b), where p, q are continuous on (a, b). By completing the following steps, prove that if ? is not identically zero, then its zeros in (a, b) are isolated, i.e., if ?(t0) = 0, then there exists a ? > 0 such that for ?(t) 0 for 0 < |t-t0| < ?.(a) Suppose ?(t0)=0 and assume to the contrary that for each n = 1,2,…, the function ? f has a zero at tn, where 0 < |t0-tn| < 1/n. Show that this implies ?’(t0)=0 [Hint: Consider the difference quotient for ? at t0.](b) With the assumptions of part (a), we have ?(t0)= ?’(t0)=0 Conclude from this that ? must be identically zero, which is a contradiction. Hence, there is some integer n0 such that ?(t) is not zero for 0 < |t-t0| < 1/n0

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Introduction to Chapter 1:  Difference of Squares: a -b =(a-b)(a+b)  Zero Product Property: ab=0; a, b=0 1.1 Linear Equations are first-degree equations in one variable that contain the equality symbol, “=”. Linear equation in the variable x can be expresses in the standard form, ax+b=c, where a ≠, b ∊ ℝ The domain is the largest set of acceptable input values. Of note, the denominator of a fraction cannot be equal to zero. 1.1.1 Solve ax+b=0, solve for x ax+(b-b)=(0-b) 1. Isolate x by subtracting b from both sides. x/a=-b/a 2. Divide both sides by a x=-b/a 1.1.2 Solve 5x-3=2x+8 -2x -2x 1. Subtract 2x from each side 3x-3=8 +3 +3 2. Ad

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##### ISBN: 9780321747730

This full solution covers the following key subjects: Zero, Where, zeros, such, identically. This expansive textbook survival guide covers 67 chapters, and 2118 solutions. The full step-by-step solution to problem: 53E from chapter: 4.7 was answered by , our top Calculus solution expert on 07/11/17, 04:37AM. This textbook survival guide was created for the textbook: Fundamentals of Differential Equations , edition: 8. Since the solution to 53E from 4.7 chapter was answered, more than 269 students have viewed the full step-by-step answer. The answer to “Isolated Zeros. Let ?(t) be a solution to y’’+py’+qy = 0 on (a, b), where p, q are continuous on (a, b). By completing the following steps, prove that if ? is not identically zero, then its zeros in (a, b) are isolated, i.e., if ?(t0) = 0, then there exists a ? > 0 such that for ?(t) 0 for 0 < |t-t0| < ?.(a) Suppose ?(t0)=0 and assume to the contrary that for each n = 1,2,…, the function ? f has a zero at tn, where 0 < |t0-tn| < 1/n. Show that this implies ?’(t0)=0 [Hint: Consider the difference quotient for ? at t0.](b) With the assumptions of part (a), we have ?(t0)= ?’(t0)=0 Conclude from this that ? must be identically zero, which is a contradiction. Hence, there is some integer n0 such that ?(t) is not zero for 0 < |t-t0| < 1/n0” is broken down into a number of easy to follow steps, and 149 words. Fundamentals of Differential Equations was written by and is associated to the ISBN: 9780321747730.

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