Show from first principles, Le., by using the definition of linear inde- pendence, that if µ = z + iy, y # 0 is an eigenvalue of a real matrix A with associated eigenvector e=u+ iw, then the two real solutions Y() = "(u cos dt – w sin be) and Z(1) = e“(usin de + w cos be) are linearly independent solutions of X = AX. (b) Use (a) to solve the system x - (; )x. NB: Real solutions are required. Hint: Recall that if the roots of C(A) = 0 occur in complex conjugate parts, then any one of the two eigenvalues yields two linearly indepen- dent solutions. The second will yleld solutions which are identical (up to a constant) to the solutions already found. Check Theorem 2.19 (Equation 2.2) and Example 2.20 on page 32 of the study guide in this regard.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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This is an ordinary differential equation question. Please help with both (a) and (b)

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(a)
Show from first principles, i.e., by using the definition of linear inde-
pendence, that if u = x+ iy, y +0 is an eigenvalue of a real matrix
A with associated eigenvector v = u+ iw, then the two real solutions
Y(t) = ea" (u cos bt – w sin bt)
and
Z(t) = e" (usin bt + w cos bt)
are linearly independent solutions of X = AX.
(b)
Use (a) to solve the system
X = ( :)
3
X.
-8
W
NB: Real solutions are required.
Hint: Recall that if the roots of C(A) = 0 occur in complex conjugate
parts, then any one of the two eigenvalues yields two linearly indepen-
dent solutions. The second will yield solutions which are identical (up
to a constant) to the solutions already found. Check Theorem 2.19
(Equation 2.2) and Example 2.20 on page 32 of the study guide in this
regard.
ll
06:53 AM
2022-03-24 Page: 1 of 1 Words: 3
B A e
E E = 709%
+
Transcribed Image Text:Document1 - Microsoft Word (Product Activation Failed) W File Home Insert Page Layout References Mailings Review View % Cut A Find - - A A =,三,“ 请 T Aal AaBbCcDc AaBbCcD AaBbC AaBbCc AaBI AqBbCcl 1 No Spaci. Heading 1 Heading 2 Calibri (Body) - 11 Aa E Copy a Replace Paste IU- abe x, x* ab A I Normal Change Styles - Select - Title Subtitle Format Painter 6. Clipboard Font Paragraph Styles Editing 2:1:1:1 1::2:1·3:1:4:15.16.17:18.1 9.1 10. 1 11. | 12.1 13.1 14:1 15: 1 17: 1 18: (a) Show from first principles, i.e., by using the definition of linear inde- pendence, that if u = x+ iy, y +0 is an eigenvalue of a real matrix A with associated eigenvector v = u+ iw, then the two real solutions Y(t) = ea" (u cos bt – w sin bt) and Z(t) = e" (usin bt + w cos bt) are linearly independent solutions of X = AX. (b) Use (a) to solve the system X = ( :) 3 X. -8 W NB: Real solutions are required. Hint: Recall that if the roots of C(A) = 0 occur in complex conjugate parts, then any one of the two eigenvalues yields two linearly indepen- dent solutions. The second will yield solutions which are identical (up to a constant) to the solutions already found. Check Theorem 2.19 (Equation 2.2) and Example 2.20 on page 32 of the study guide in this regard. ll 06:53 AM 2022-03-24 Page: 1 of 1 Words: 3 B A e E E = 709% +
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