Query 29 Differential Equations

Question:  Find the propagator matrix Phi(t) = e^(t A) for the system

y ' = [5, -4; 5, -4] y

and use to find y(3) given that y(1) == [ 1, 0].

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Question:  Use a propagator matrix to find y(1), given  y ( 0 ) = [1; 1; 0], for the system

y ' = [ 1, 1, 1; 0, 2, 1; 0, 0, -1 ] y.

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Given Solution:  Given the solution matrix

psi(t) = [t, t^2; 1, 2 t]

find the propagator matrix phi(t, s), t > 0, s > 0.

Is the propagator matrix a function of t - s?

Find y(3) given that y(1) = [1, -1].

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Question:  Suppose that T^-1 A T = [lambda_1, 0; 0, lambda_2].

Let p(A) be the matrix polynomial 2 A^3 - A + 3 I.

Find the matrix B such that p(A) = T B T^-1.

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Question:  Let A be invertible and diagonalizable, and let D = T^-1 A T be the matrix as diagonalized by a similarity transform.

Show that D is invertible.

Show that A^-1 is diagonalizable by the same similarity transform that diagonalizes A (i.e., show that D^-1 = T^-1 A^-1 T).

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Question:  Let A be a diagonalizable 2 x 2 matrix with

lambda_1 = 1/4, x_1 = [2, 5]

and

lambda_2 = 1/2, x_2 = [1, 3].

Find cos(pi A) and sin(pi A).

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Question:  Let A be a diagonalizable 2 x 2 matrix with

lambda_1 = 1/4, x_1 = [2, 5]

and

lambda_2 = 1/2, x_2 = [1, 3].

Solve the equation

y '' + y ' + A y = 0.

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