general solution
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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Need help for #17 and 19 please
![The image contains a series of problems related to systems of linear differential equations and their solutions using eigenvalue methods.
**Problems:**
15.
\[ x'_1 = 7x_1 - 5x_2, \quad x'_2 = 4x_1 + 3x_2 \]
16.
\[ x'_1 = -50x_1 + 20x_2, \quad x'_2 = 100x_1 - 60x_2 \]
In Problems 17 through 25, determine the eigenvalues of the coefficient matrix by inspection and factoring. Use the eigenvalue method to find a general solution for each system.
17.
\[ x'_1 = 4x_1 + 2x_2 + 4x_3, \quad x'_2 = x_1 + 7x_2 + x_3, \quad x'_3 = 4x_1 + x_2 + 4x_3 \]
18.
\[ x'_1 = x_1 + 2x_2 + 2x_3, \quad x'_2 = 2x_1 + 7x_2 + x_3, \quad x'_3 = 2x_1 + 2x_2 + 7x_3 \]
19.
\[ x'_1 = 4x_1 + x_2 + x_3, \quad x'_2 = -x_1 + 4x_2 + x_3, \quad x'_3 = x_1 + x_2 + 4x_3 \]
20.
\[ x'_1 = 5x_1 + x_2 + 3x_3, \quad x'_2 = -x_1 + 7x_2 + x_3, \quad x'_3 = 3x_1 + x_2 + 5x_3 \]
21.
\[ x'_1 = 5x_1 - 6x_3, \quad x'_2 = 2x_1 - x_2 - 2x_3, \quad x'_3 = 4x_1 - 2x_2 - 4x_3 \]
22.
\[ x'_1 = 3x_1 + 2x_2 + 2x_3](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F03106fa3-330c-422f-ad56-a6718a5c3a13%2F76d37a4d-5c54-46dd-b623-734a944c7f84%2Fwxqmujs.jpeg&w=3840&q=75)
Transcribed Image Text:The image contains a series of problems related to systems of linear differential equations and their solutions using eigenvalue methods.
**Problems:**
15.
\[ x'_1 = 7x_1 - 5x_2, \quad x'_2 = 4x_1 + 3x_2 \]
16.
\[ x'_1 = -50x_1 + 20x_2, \quad x'_2 = 100x_1 - 60x_2 \]
In Problems 17 through 25, determine the eigenvalues of the coefficient matrix by inspection and factoring. Use the eigenvalue method to find a general solution for each system.
17.
\[ x'_1 = 4x_1 + 2x_2 + 4x_3, \quad x'_2 = x_1 + 7x_2 + x_3, \quad x'_3 = 4x_1 + x_2 + 4x_3 \]
18.
\[ x'_1 = x_1 + 2x_2 + 2x_3, \quad x'_2 = 2x_1 + 7x_2 + x_3, \quad x'_3 = 2x_1 + 2x_2 + 7x_3 \]
19.
\[ x'_1 = 4x_1 + x_2 + x_3, \quad x'_2 = -x_1 + 4x_2 + x_3, \quad x'_3 = x_1 + x_2 + 4x_3 \]
20.
\[ x'_1 = 5x_1 + x_2 + 3x_3, \quad x'_2 = -x_1 + 7x_2 + x_3, \quad x'_3 = 3x_1 + x_2 + 5x_3 \]
21.
\[ x'_1 = 5x_1 - 6x_3, \quad x'_2 = 2x_1 - x_2 - 2x_3, \quad x'_3 = 4x_1 - 2x_2 - 4x_3 \]
22.
\[ x'_1 = 3x_1 + 2x_2 + 2x_3
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