Query 27 Differential Equations

Question:  Find the eigenvalues of the matrix [3, 1; -2, 1] and find the corresponding eigenvectors.

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Question:  Suppose that i + 1 is an eigenvalue of a matrix A and [-1 + i, i ] is a corresponding eigenvector.  Find a fundamental set of real solutions to the equation y ' = A y.

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Question:  Solve the equation

y ' = [0, -9; 1, 0] y

with initial condition

y(0) = [6, 2].

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Question:  Find all values of mu such that any fundamental set [ y_1, y_2 ] of the system

y ' = [1, 3; mu, -2] y

has the property that the limit of the expression (y_1(t))^2 + (y_2(t))^2, as t -> infinity, is zero.

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Question:  A particle moves in an unspecified force field in such a way that its position vector r(t) = x(t) i + y(t) j and the corresponding velocity vector v(t) = r ' (t) satisfy the equation

v ' = 2 k X v

Write this condition as a system

v ' = A v,

with v = [v_x; v_y].

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A particle moves in an unspecified force field in such a way that its position vector r(t) = x(t) i + y(t) j and the corresponding velocity vector v(t) = r ' (t) satisfy the equation

v ' = 2 k X v

Write this condition as a system

v ' = A v,

with v = [v_x; v_y].

Self-critique (if necessary):

Self-critique rating: