1) Consider the following linear differential equations:
a. y'' + 2y' - 3y = 0 y(0) = 1, y'(0) = 0
b. 2y'' - 5y' + 3y = 0 y(0) = 1, y'(0) = 1
c. y'' + 5y' - 6y = 0 y(0) = 1, y'(0) = -1
Perform the following tasks on each of the above:
i. Write the differential equation as a system of first order differential equations expressed in matrix form.
ii. For each system, find the equilibrium points.
iii. Find the solution to the homogeneous differential equation using the eigenvalues and eigenvectors of the system matrix.
2) Find the solution to the following dynamical system equations given below. Assume that
Also, determine the nature of the equilibrium point based on the eigenvalues and eigenvectors of the system
3) Solve the following non-homogenous dynamical system equations using Laplace transform.
Recall that for:
x'(t) = Ax(t) + Bu(t)
X(x) = (sI - A)-1 {x(0) + BU(s)}
4) Fill the following table of eigenvalue response type for linear autonomous homogenous second order system. Identify whether or not the system represented is a stable system name the type of equilibrium point according to: stable/unstable, node/spiral center saddle, etc. Verify your solution using PPLANE8 software.
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Matrix and Eigenvalue Condition
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Name of Response
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A = [2 0; 0 6]; λ1 = -1 ; λ2 = -2
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A = [-2 - 2; 0 -6]; λ1 = -2 ; λ2 = -6
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A = [3 -2; 4 -1]; λ1 = 1 + 2i ; λ2 = 1 - 2i
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A = [-3 2; -4 1]; λ1 = -1 + 2i ; λ2 = - 1 - 2i
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A = [-1 - 6; -6 0]; λ1 = 2 ; λ2 = -3
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A = [0 -1; 1 0]; λ1 = i ; λ2 = -i
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