Find the general solution of y " - 7y " + 16y' – 12y =0 given that r = 3 is a root of the characteristic equation.
Find the general solution of y " - 7y " + 16y' – 12y =0 given that r = 3 is a root of the characteristic equation.
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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![**Find the General Solution**
Given the differential equation:
\[ y''' - 7y'' + 16y' - 12y = 0 \]
with the root \( r_1 = 3 \) of the characteristic equation, determine the general solution.
**Options:**
a) \( y = C_1 e^{2x} + C_2 e^{-2x} + C_3 e^{3x} \)
b) \( y = C_1 e^{-2x} + C_2 x e^{-2x} + C_3 e^{-3x} \)
c) \( y = C_1 e^{2x} + C_2 x e^{2x} + C_3 e^{-3x} \)
d) \( y = C_1 e^{2x} + C_2 x e^{2x} + C_3 e^{3x} \)
e) \( y = C_1 e^{2x} + C_2 e^{3x} + C_3 x e^{3x} \)
f) None of the above.
**Notes:**
- \( C_1, C_2, \) and \( C_3 \) are constants.
- Evaluate the characteristic equation and compute possible solutions based on given roots.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F88afae02-80d5-49da-ad82-a7933e6f4445%2F14d55f35-f835-4839-8418-debc1570bd09%2Fh43kjq8_processed.png&w=3840&q=75)
Transcribed Image Text:**Find the General Solution**
Given the differential equation:
\[ y''' - 7y'' + 16y' - 12y = 0 \]
with the root \( r_1 = 3 \) of the characteristic equation, determine the general solution.
**Options:**
a) \( y = C_1 e^{2x} + C_2 e^{-2x} + C_3 e^{3x} \)
b) \( y = C_1 e^{-2x} + C_2 x e^{-2x} + C_3 e^{-3x} \)
c) \( y = C_1 e^{2x} + C_2 x e^{2x} + C_3 e^{-3x} \)
d) \( y = C_1 e^{2x} + C_2 x e^{2x} + C_3 e^{3x} \)
e) \( y = C_1 e^{2x} + C_2 e^{3x} + C_3 x e^{3x} \)
f) None of the above.
**Notes:**
- \( C_1, C_2, \) and \( C_3 \) are constants.
- Evaluate the characteristic equation and compute possible solutions based on given roots.
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