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(a) Show from first principles, i.e., 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 v = u + iw, then the two real solutions Y(t) = e“ (u cos bt – w sin bt) and Z(1) = e"(usin ôt + w cos bt) are linearly independent solutions of X = AX. (b) Use (a) to solve the system x- (: )x 3 X. NB: Real solutions are required.

(a) Show from first principles, i.e., 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 v = u + iw, then the two real solutions Y(t) = e“ (u cos bt – w sin bt) and Z(1) = e"(usin ôt + w cos bt) are linearly independent solutions of X = AX. (b) Use (a) to solve the system x- (: )x 3 X. NB: Real solutions are required.

Advanced Engineering Mathematics
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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W Document1 - Microsoft Word (Product Activation Failed) File Home Insert Page Layout References Mailings Review View a ? % Cut A Find - Calibri (Body) - 14 - A A Aa Aal AaBbCcDc AaBbCcD AaBbC AaBbCc AaBI AqBbCcl E Copy ag Replace BI U ab A I Normal Paste - * abe x, x A I No Spaci.. Heading 1 Heading 2 Title Subtitle Change W Format Painter Styles - Select - Clipboard Font Paragraph Styles Editing 2:1·1•I 11• 2:1:3:1·4: 15.1 6:1 7:18•1 9.1 10. 11· I·12: 1 •13. 1 14: I 15. 1 : 17: 1 18. ORDINARY DIFFERENTIAL EQUATIONS PLEASE ANSWER ALL QUESTIONS (a) Show from first principles, i.e., by using the definition of linear inde- pendence, that if µ = 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) = eat (u cos bt – w sin bt) and Z(t) = e"(u sin bt + w cos bt) are linearly independent solutions of X = AX. (b) Use (a) to solve the system X = -8 X. NB: Real solutions are required. 07:14 PM 2022-05-18 Page: 1 of 1 Words: 9 昆 昌 90% +
Transcribed Image Text:W Document1 - Microsoft Word (Product Activation Failed) File Home Insert Page Layout References Mailings Review View a ? % Cut A Find - Calibri (Body) - 14 - A A Aa Aal AaBbCcDc AaBbCcD AaBbC AaBbCc AaBI AqBbCcl E Copy ag Replace BI U ab A I Normal Paste - * abe x, x A I No Spaci.. Heading 1 Heading 2 Title Subtitle Change W Format Painter Styles - Select - Clipboard Font Paragraph Styles Editing 2:1·1•I 11• 2:1:3:1·4: 15.1 6:1 7:18•1 9.1 10. 11· I·12: 1 •13. 1 14: I 15. 1 : 17: 1 18. ORDINARY DIFFERENTIAL EQUATIONS PLEASE ANSWER ALL QUESTIONS (a) Show from first principles, i.e., by using the definition of linear inde- pendence, that if µ = 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) = eat (u cos bt – w sin bt) and Z(t) = e"(u sin bt + w cos bt) are linearly independent solutions of X = AX. (b) Use (a) to solve the system X = -8 X. NB: Real solutions are required. 07:14 PM 2022-05-18 Page: 1 of 1 Words: 9 昆 昌 90% +
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