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(e) Use your answer to part (a) to find the general solution of the following system of linear differential equations, writing the solution as an equation for x and an equation for y: * = ý 6x + 4y -2x - 3y

(e) Use your answer to part (a) to find the general solution of the following system of linear differential equations, writing the solution as an equation for x and an equation for y: * = ý 6x + 4y -2x - 3y

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.4: Similarity And Diagonalization
Problem 6EQ
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question 8. The given matrix is A² (603) 4 -2 - A and for each a) Here we have to find the eigen Valves of eigen Valve we have to find a Corresponding eigen rector in the form of , where a is a Positive number. eigen Value of Athen Let & be an LA-X 1²0 6- X 4 ㅕ -0 -2 -3) =1 (6-X) (-3-X) + 8 = 0 =1-18-6 X + 3 + ³ + 8 = 0 -3 -10 = 0 -1 =1 X-SX 72 X-10=0 = > >=-5) +2(X - 5) = 0 = (x+2)(x-5) = 0 کر 2-= X Take AF-2 and 1₂ s let V= Then (A-1₁I) V-6 7(A +2₁) V=0 [Xi = -2] = 6 + ² 4 2 - 2 [V₂] be an eigen vector of A Corresponding to di= -2 - 3+2 V₁ √₂/ 0 8 V₁T Y√₂ = 0 and 2V₁ V₂ =0 + 2V, ond - 2V₁ + V₂ =6 ㅋ = 2V₁ + V₂ =0 V₂=0 8 12 V₁ = 1 and V₂ = 2 > as on We Solving, get Thus we can take V₁ = (-2) √₂
Transcribed Image Text:question 8. The given matrix is A² (603) 4 -2 - A and for each a) Here we have to find the eigen Valves of eigen Valve we have to find a Corresponding eigen rector in the form of , where a is a Positive number. eigen Value of Athen Let & be an LA-X 1²0 6- X 4 ㅕ -0 -2 -3) =1 (6-X) (-3-X) + 8 = 0 =1-18-6 X + 3 + ³ + 8 = 0 -3 -10 = 0 -1 =1 X-SX 72 X-10=0 = > >=-5) +2(X - 5) = 0 = (x+2)(x-5) = 0 کر 2-= X Take AF-2 and 1₂ s let V= Then (A-1₁I) V-6 7(A +2₁) V=0 [Xi = -2] = 6 + ² 4 2 - 2 [V₂] be an eigen vector of A Corresponding to di= -2 - 3+2 V₁ √₂/ 0 8 V₁T Y√₂ = 0 and 2V₁ V₂ =0 + 2V, ond - 2V₁ + V₂ =6 ㅋ = 2V₁ + V₂ =0 V₂=0 8 12 V₁ = 1 and V₂ = 2 > as on We Solving, get Thus we can take V₁ = (-2) √₂
148) an eigen vector of a Let x = (2₂) bc (A-XI)x=0 X 2 = S to There fore Vector is A Corresponding to A X1 = -2 eigen = (A -51) X = 0 [°° >₂=5] (625 $35) (2) (8) =1 -1 (1-2-3) (62) -(8 =1 x₁ +4x₂ = 0 and -22₁ -8 2₂=0 =1 IC₁ + 4x₂ = 0 and x₁ 4x2 = 0 =1 хсi + Чх 2 : 0 On solving, we get 20₁=4 and 2² = 1 Thus we Can take V₂ = (-1) as an Vector of A Corresponding to x = 6x + 4y y = -2x - 3y on X₁ = -2 is an eigen Value of A V2 > answer above. eigen Vector 25 Then of Corresponding (e) Use your answer to part (a) to find the general solution of the following system of linear differential equations, writing the solution as an equation for x and an equation for y: and Corresponding eigen
Transcribed Image Text:148) an eigen vector of a Let x = (2₂) bc (A-XI)x=0 X 2 = S to There fore Vector is A Corresponding to A X1 = -2 eigen = (A -51) X = 0 [°° >₂=5] (625 $35) (2) (8) =1 -1 (1-2-3) (62) -(8 =1 x₁ +4x₂ = 0 and -22₁ -8 2₂=0 =1 IC₁ + 4x₂ = 0 and x₁ 4x2 = 0 =1 хсi + Чх 2 : 0 On solving, we get 20₁=4 and 2² = 1 Thus we Can take V₂ = (-1) as an Vector of A Corresponding to x = 6x + 4y y = -2x - 3y on X₁ = -2 is an eigen Value of A V2 > answer above. eigen Vector 25 Then of Corresponding (e) Use your answer to part (a) to find the general solution of the following system of linear differential equations, writing the solution as an equation for x and an equation for y: and Corresponding eigen
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