Find the general solution of у "— 2у "— 16у'+ 32у -0 given that r = 2 is a root of the characteristic equation. 4x -4x -4x b) O y=C, e*+Ce 4 x c) O y=C,e*+C, 4x d) O y-c, + -4x C, e 2 4x + C,e + C, f) O None of the above.
Find the general solution of у "— 2у "— 16у'+ 32у -0 given that r = 2 is a root of the characteristic equation. 4x -4x -4x b) O y=C, e*+Ce 4 x c) O y=C,e*+C, 4x d) O y-c, + -4x C, e 2 4x + C,e + C, f) O None of the above.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![**Problem: Find the General Solution**
Find the general solution of the differential equation given that \( r_1 = 2 \) is a root of the characteristic equation.
\[
y''' - 2y'' - 16y' + 32y = 0
\]
**Options:**
a) \( y = C_1 e^{2x} + C_2 e^{4x} + C_3 x e^{4x} \)
b) \( y = C_1 e^{-2x} + C_2 e^{4x} + C_3 e^{-4x} \)
c) \( y = C_1 e^{-2x} + C_2 e^{-4x} + C_3 e^{4x} \)
d) \( y = C_1 e^{2x} + C_2 e^{4x} + C_3 e^{-4x} \)
e) \( y = C_1 e^{2x} + C_2 e^{4x} + C_3 e^{4x} \)
f) \(\circ\) None of the above.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F88afae02-80d5-49da-ad82-a7933e6f4445%2F90d4c0c1-acd3-4bde-b024-080e79d82de2%2Fkf6qsj_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem: Find the General Solution**
Find the general solution of the differential equation given that \( r_1 = 2 \) is a root of the characteristic equation.
\[
y''' - 2y'' - 16y' + 32y = 0
\]
**Options:**
a) \( y = C_1 e^{2x} + C_2 e^{4x} + C_3 x e^{4x} \)
b) \( y = C_1 e^{-2x} + C_2 e^{4x} + C_3 e^{-4x} \)
c) \( y = C_1 e^{-2x} + C_2 e^{-4x} + C_3 e^{4x} \)
d) \( y = C_1 e^{2x} + C_2 e^{4x} + C_3 e^{-4x} \)
e) \( y = C_1 e^{2x} + C_2 e^{4x} + C_3 e^{4x} \)
f) \(\circ\) None of the above.
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