Explanation Procedure used: Solution to a second order differential equation. For a second order differential equation a y ″ + b y ′ + c y = 0 , where a , b and c are constants, the auxiliary equation will be of the form a m 2 + b m + c = 0 . There will be three forms of the general solution of a y ″ + b y ′ + c y = 0 corresponding to three cases which dependent to the solution of auxiliary equation. 1. m 1 and m 2 are real and distinct roots that is possible when ( b 2 − 4 a c > 0 ) 2. m 1 and m 2 are real and equal roots that is possible when ( b 2 − 4 a c = 0 ) 3. m 1 and m 2 are conjugate complex roots ( b 2 − 4 a c < 0 ) Case I: Distinct Real Roots The general solution of a y ″ + b y ′ + c y = 0 on the interval ( − ∞ , ∞ ) will be y = c 1 e m 1 x + c 2 e m 2 x . Case II: Repeated Real Roots The general solution will be y = c 1 e m 1 x + c 2 x e m 1 x . Case III: Conjugate Complex Roots If m 1 and m 2 are complex roots, then m 1 = α + i β and m 2 = α − i β , where α and β > 0 are real and i 2 = − 1 . The general solution will be y = c 1 e a x cos β x + c 2 e a x sin β x = e a x ( c 1 cos β x + c 2 sin β x ) Calculation: Consider the given differential equation y ″ + 8 y ′ + 16 y = 0

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Differential Equations

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Chapter 3.3, Problem 5E

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[) Hwk 25 Hwk 28 - (MA 244-03) (SP25) || X Success Confirmation of Questic X + https://www.webassign.net/web/Student/Assignment-Responses/submit?dep=36606607&tags=autosave#question 384855 DETAILS MY NOTES LARLINALG8 7.2.001. 1. [-/2.85 Points] Consider the following. -14 60 A = [ -4-5 P = -3 13 -1 -1 (a) Verify that A is diagonalizable by computing P-1AP. P-1AP = 具首 (b) Use the result of part (a) and the theorem below to find the eigenvalues of A. Similar Matrices Have the Same Eigenvalues If A and B are similar n x n matrices, then they have the same eigenvalues. (11, 12) = Need Help? Read It SUBMIT ANSWER 2. [-/2.85 Points] DETAILS MY NOTES LARLINALG8 7.2.007. For the matrix A, find (if possible) a nonsingular matrix P such that P-1AP is diagonal. (If not possible, enter IMPOSSIBLE.) P = A = 12 -3 -4 1 Verify that P-1AP is a diagonal matrix with the eigenvalues on the main diagonal. P-1AP = Need Help? Read It Watch It SUBMIT ANSWED 80% ill จ ASK YOUR TEACHER PRACTICE ANOTHER ASK YOUR…

[) Hwk 25 → C Hwk 27 - (MA 244-03) (SP25) IN X Answered: [) Hwk 25 4. [-/4 Poir X + https://www.webassign.net/web/Student/Assignment-Responses/submit?dep=36606606&tags=autosave#question3706544_6 3. [-/2.85 Points] DETAILS MY NOTES LARLINALG8 7.1.021. Find the characteristic equation and the eigenvalues (and a basis for each of the corresponding eigenspaces) of the matrix. 2 -2 5 0 3 -2 0-1 2 (a) the characteristic equation (b) the eigenvalues (Enter your answers from smallest to largest.) (1, 2, 13) = ·( ) a basis for each of the corresponding eigenspaces X1 x2 = x3 = Need Help? Read It Watch It SUBMIT ANSWER 4. [-/2.85 Points] DETAILS MY NOTES LARLINALG8 7.1.041. Find the eigenvalues of the triangular or diagonal matrix. (Enter your answers as a comma-separated list.) λ= 1 0 1 045 002 Need Help? Read It ASK YOUR TEACHER PRACTICE ANOTHER ASK YOUR TEACHER PRACTICE ANOTHER ill

[) Hwk 25 4. [-/4 Points] Hwk 25 - (MA 244-03) (SP25) || X Answered: Homework#7 | bartle X + https://www.webassign.net/web/Student/Assignment-Responses/last?dep=36606604 DETAILS MY NOTES LARLINALG8 6.4.019. Use the matrix P to determine if the matrices A and A' are similar. -1 -1 12 9 '-[ ¯ ¯ ], ^ - [ _—2—2 _ ' ], ^' - [ ˜³ −10] P = 1 2 A = -20-11 A' -3-10 6 4 P-1 = Are they similar? Yes, they are similar. No, they are not similar. Need Help? Read It SUBMIT ANSWER P-1AP = 5. [-/4 Points] DETAILS MY NOTES LARLINALG8 6.4.023. Suppose A is the matrix for T: R³ - → R³ relative to the standard basis. Find the diagonal matrix A' for T relative to the basis B'. A' = -1 -2 0 A = -1 0 0 ' 0 02 B' = {(−1, 1, 0), (2, 1, 0), (0, 0, 1)} ☐☐☐ ↓ ↑ Need Help? Read It Update available →] - restart now ASK YOUR T Sync and save data { Sign In ill ↑ New tab HT New window N New private window +HP ASK YOUR T Bookmarks History Downloads > > HJ Passwords Add-ons and themes HA Print... HP Save page as... HS…

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The general solution of the second order differential equation y ″ + 8 y ′ + 16 y = 0 .

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