Let x(t) =x₁ (t)x'₂ (t)=If x (0)[x₂(t): =x₁(t)x₂ (t)be a solution to the system of differential equations:5B5Put the eigenvalues in ascending order when you enter x1(t), 2 (t) below.x₁(t):Jexp(t)+Jexp(t)+4x₁(t) + 2x₂ (t)-3x1(t) + 9x2(t)find x (t).exp([exp(t)exp(t)

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Answered: Let x(t) = x₁ (t) x'₂ (t) = If x (0) […

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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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 differential equation: y(5) + 4y(4) - y''' - 14y'' + 10y' = 0 Determine the characteristic equation Φ(m) = 0 associated to the differential equation. Completely factor the characteristic polynomial Φ(m). Determine the solution set of the characteristic equation Φ(m) = 0. For each of the solutions must indicate their multiplicity.
Suppose tx1 + 2x2 + sec(t), sin(t)x1+ tx2 – 3. This system of linear differential equations can be put in the form æ' = P(t)æ + g(t). Determine P(t) and g(t). P(t) = g(t) =
3. Consider the system of differential equations x' (t) = Ax(t) for t > 0, x(0) = (b) where A € C²x2 and a : R+ → C². Solve (1) by using the matrix exponential, when (a) A A = (1), H₁ 3 1 (1)
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 =
2. Consider the eigenvalue problem, y" + Ay = 0, for 0 < x < 7/2; y' (0) = 0, y(n/2) = 0. Find all eigenvalues and eigenfunctions for the case where the auxiliary equation has complex roots.
Transfer the following differential equation: - 4t y" + 3y" + 2y' — 5y = t² + e²¯ into a system of first order differential equations in the form of x' Ax + ģ(t) What are , A, and ģ(t)? = x(t) x (t) x (t) x (t) = = = = x₁ (t) x₂ (t) x3 (t). x₁ (t) x₂ (t) x3 (t). [T₁ (t)] x₂ (t) x3 (t). X1 x₁ (t) = = = Y Y D C Y x₂ (t) - = x3 (t). Y y' 2 " A = A - A = cos t A = 0 5 1 0 5 1 0 0 1 -2 -3 0 1 0 EH -5 2 3 0 0 1 1 0 1 -2 -3 -3 1 0 0 1 -2 5 " 0 0 1 g(t) 2 2 g(t) " g(t) = g(t) = = e -4t 0 0 +e-4t cos(t). + e t² 0 0 -4t cos(t) cos(t). 0 0 t² + e-4t cos(t).
Consider the linear homogeneous second order ODE 3y''-11y'-4y=0 A) Write out the characteristic equation corresponding to the differential equation. B) Solve the characteristic equation and write out the general solution to the equation. (solve by factoring, completing the square, or use the quadratic equation)

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