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(D³ + D² – 21D – 45)y = 0

(D³ + D² – 21D – 45)y = 0

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section: Chapter Questions
Problem 5T
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Answer the following homogeneous linear differential equations with constant coefficients.

3. (D³ + D² – 21D – 45)y = 0 %3D
Transcribed Image Text:3. (D³ + D² – 21D – 45)y = 0 %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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