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list.) D³ 18D2+81D DD-9 X

list.) D³ 18D2+81D DD-9 X

Advanced Engineering Mathematics
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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Please answer the two questions ty!

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
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))
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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