7. Write the 4th order ordinary differential equation as a matrix system of first order ordinarydifferential equations with an initial vector condition.a) In the form x' = Ax +g(t) with x(1) = xo & indicate what the terms A, x, xo, and g(t) are.b) If the Wronskian at t = 1 is 5daydt4-d³ ydt 3td³ydt 3(1) = 1t²d²ydt²d²ydt²-t3(1) = 1dydtdydtt4y = 7 cos (t)(1) = 2 y(1) = 9

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Answered: 7. Write the 4th order ordinary…

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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Express the system of differential equations in matrix notation. - x" -6x+ty (sint)x = 0 y'"'+5y"-tx' + 3y' + e¯¹x=0 Which of the following sets of definitions allows the given system to be written as an equivalent system in normal form using only the new variables? OA. x₁ = x, x₂ = x², x3 = y, x4 = y' OB. X₁ =x', x=x", x3 = y', x4 = y", x5 =y"" OC. x₁ = x, x2 =x', x3 = y, x4 = y', x5 =y" OD. x₁ = x, x₂ =x", x3 = y, x4 = y'"' Write the system of equations using matrix notation. Select the correct choice below and fill in the answer box to complete your choice. (Type an exact answer.) x1 X1 x2 X3 ○ A. ха X5 X5 ہیں ○ B. کا 5 x1 x2 X3 ха
Consider a system of two linear first-order ordinary differential equations: y₁ = y₁ - y2, y₂ = 2y1 - y2 - a) The corresponding Oλ₁ = 1, №₂ = −1 b) The corresponding OIII and IV where 1:4₁ = II:42 = 1+ 2 (¹+4) (¹3²) 1 - 2i 2(1 + i) (2(1²-+-+)) i) III:u1 IV:4₂ = eigenvalues are = OA₁ = 1 + 1, A₂ = 1 - i OA₁i, A₂ = −i = eigenvectors of this linear ODE system are: Oll and III OI and IV OI and II c) The phase portrait for this system of ODEs is Stable focus with spiral in Centre Unstable focus with spiral out Stable node
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Solve the first order nonhomogeneous system of differential equations
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1) Find the general solution to the following differential equation. dy +y (cotx) = 5e* dx 2) Reduce the following differential equation to first-order and solve it. (1 - x2)y" – 2xy' + 2y = 0 ; Y1 = x 3) Let A be the augmented matrix of a linear system. Find the solution for this linear system. [1 2 3 01 A = 1 1 1 : 0 L5 7 9 o] ALTIN DAS
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3. Consider the system of two differential equations given by x' = Ax, which has the [v₁ + iw₁] complex eigenvalue A = a+iß with the corresponding eigenvector v = V2+iW₂ Find the (real-valued) general solution of the system. Show all your work.
Consider the second-order system of differential equations d²x dt² d²y dt2 =x+3y, The coefficient matrix 1 3 3 3x+y. wwwwww has eigenvectors Hand [1] are 4 and - 2. Select the option that gives the general solution of this system. Select one: ° 1) = 0[¹] con (24) + C₁ [₁] (²0₁₁+₂ [1¹] CA cos(2t) sin(2t) + C3 ev2t [*] = C₁ [1] cos(2t) + 0₂ sin(2t) + C3 cos(√2t) + C4 Casin(√21) = + HHH] co √21) + C [¹] sin (√2) C₁ e2t cos(√2t) C4 C3 and the corresponding eigenvalues [3] = C₁ [1] cos(2t) + C₂ e2t 2t [1] = ₂ [1] ² + ₂ [1] ² ² + ₁ [₁¹] e√²+ + C₁ [ 1¹ ] e-√₂² 2t e C4 A WEW MEXAND sin(2t) + C3 · [₁¹] ₁ √²+ + 0₁ [1¹] ₁ = √² √√2t e
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1. Find real-valued fundamental solutions of the differential equation x' = Ax, where matrix A has eigenvalues and eigenvectors - [₁4] - 6.2) A₁ = −2+i V₁ = A2 = -2-i V2 =

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