12. (D³ – 4D² + 29D)y = 0 ; y(0) = 0, y'(0) = 1,y"(0) = -1 )

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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Answer the following homogeneous linear differential equations with constant coefficients.

12. (D³ – 4D² + 29D)y = 0 ; y(0) = 0, y'(0) = 1, y"(0) = -1
%3D
Transcribed Image Text:12. (D³ – 4D² + 29D)y = 0 ; y(0) = 0, y'(0) = 1, y"(0) = -1 %3D
Method of Solution
1. Determine the auxiliary equation f (m) = 0 from the given f(D)y = 0.
2. Factor out f(m) = 0 to get the roots m1, m2, m3, ...,Mn. Apply the necessary method in
factoring including synthetic division and quadratic formula.
3. Classify each root whether distinct, repeated or imaginary.
4. Formulate the general solution with terms according to the classification of each root m:
Distinct: Cremk*
for each root mä
Repeated: em* (cn-1x"-1
Cn-2x"-2 + ... + c,x + co) for roots m repeated n times
Imaginary: eax (c cos bx + c2 sin bx)
ах
for roots m = a ± bi
5. If indicated, apply the initial condition to get the values of c1, c2, C3, ... , Cn.
Transcribed Image Text:Method of Solution 1. Determine the auxiliary equation f (m) = 0 from the given f(D)y = 0. 2. Factor out f(m) = 0 to get the roots m1, m2, m3, ...,Mn. Apply the necessary method in factoring including synthetic division and quadratic formula. 3. Classify each root whether distinct, repeated or imaginary. 4. Formulate the general solution with terms according to the classification of each root m: Distinct: Cremk* for each root mä Repeated: em* (cn-1x"-1 Cn-2x"-2 + ... + c,x + co) for roots m repeated n times Imaginary: eax (c cos bx + c2 sin bx) ах for roots m = a ± bi 5. If indicated, apply the initial condition to get the values of c1, c2, C3, ... , Cn.
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