Find the homogeneous equation with constant coefficients that has the given general solution -2x -2x y=C, e*+C,xe *+ C, cos(2 x) + C, sin(2 x) у "+8 у'+4 у"+16у +16%3D0 b) O y4) + 2 y - 2 y'+y=0 +8 y +4 y + 16 y'+ 16y =0 d) O y4) +8 y -4 у " — 16 у'+ 16у%3D0 = -4 y + 16 у'— 16у-0 f) O None of the above.
Find the homogeneous equation with constant coefficients that has the given general solution -2x -2x y=C, e*+C,xe *+ C, cos(2 x) + C, sin(2 x) у "+8 у'+4 у"+16у +16%3D0 b) O y4) + 2 y - 2 y'+y=0 +8 y +4 y + 16 y'+ 16y =0 d) O y4) +8 y -4 у " — 16 у'+ 16у%3D0 = -4 y + 16 у'— 16у-0 f) O None of the above.
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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![**Problem Statement:**
Find the homogeneous equation with constant coefficients that has the given general solution:
\[ y = C_1 e^{-2x} + C_2 x e^{-2x} + C_3 \cos(2x) + C_4 \sin(2x) \]
**Options:**
a) \( y^{(4)} + 8 \, y''' + 4 \, y'' + 16 \, y' + 16y = 0 \)
b) \( y^{(4)} + 2 \, y''' - 2 \, y' + y = 0 \)
c) \( y^{(4)} + 8 \, y''' + 4 \, y'' + 16 \, y' + 16y = 0 \)
d) \( y^{(4)} + 8 \, y''' - 4 \, y'' - 16 \, y' + 16y = 0 \)
e) \( y^{(4)} - 4 \, y''' + 16 \, y' - 16y = 0 \)
f) None of the above.
(Note: Options a and c are identical.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F88afae02-80d5-49da-ad82-a7933e6f4445%2F08b4e29e-eb53-4af5-81bd-aa327db75c79%2Fxneql4_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the homogeneous equation with constant coefficients that has the given general solution:
\[ y = C_1 e^{-2x} + C_2 x e^{-2x} + C_3 \cos(2x) + C_4 \sin(2x) \]
**Options:**
a) \( y^{(4)} + 8 \, y''' + 4 \, y'' + 16 \, y' + 16y = 0 \)
b) \( y^{(4)} + 2 \, y''' - 2 \, y' + y = 0 \)
c) \( y^{(4)} + 8 \, y''' + 4 \, y'' + 16 \, y' + 16y = 0 \)
d) \( y^{(4)} + 8 \, y''' - 4 \, y'' - 16 \, y' + 16y = 0 \)
e) \( y^{(4)} - 4 \, y''' + 16 \, y' - 16y = 0 \)
f) None of the above.
(Note: Options a and c are identical.)
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