list.) D³ 18D2+81D DD-9 X
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Please answer the two questions ty!
Transcribed Image Text:Find linearly independent functions that are annihilated by the given differential operator. (Give as many functions as possible. Use x as the independent variable. Enter your answers as a comma-separated
list.)
D3 18D² +81D
D D-9
2
X
![Step 2
We have found that the roofs of the auxiliary equation are m₁ = 4 + 2i, and m₂ = 4 - 2i. We have been given a second-order differential equation and therefore a quadratic auxiliary equation. We know,
as in equation (8) of section 4.3, that in the case where there are conjugate complex roots a ± ßi, where a and ß> 0 are real, the solution of the homogeneous equation is
y = eax (c₁ cos(x) + C₂ sin(x)).
Therefore, for our nonhomogeneous equation, the complementary function is as follows.
Y₁ = e4x(c₁ cos(( [
])x) + C₂ sin(( [
D)x))](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc962edc4-7a48-4156-81ee-04d2df56a76e%2Fdff5e91f-4a14-4033-9437-f31c023025d9%2Fs0xro3s_processed.png&w=3840&q=75)
Transcribed Image Text:Step 2
We have found that the roofs of the auxiliary equation are m₁ = 4 + 2i, and m₂ = 4 - 2i. We have been given a second-order differential equation and therefore a quadratic auxiliary equation. We know,
as in equation (8) of section 4.3, that in the case where there are conjugate complex roots a ± ßi, where a and ß> 0 are real, the solution of the homogeneous equation is
y = eax (c₁ cos(x) + C₂ sin(x)).
Therefore, for our nonhomogeneous equation, the complementary function is as follows.
Y₁ = e4x(c₁ cos(( [
])x) + C₂ sin(( [
D)x))
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