Answer to Question #343236 in Differential Equations for Meera

Question

1a. Show from first principles, i.e., by using the definition of linear independence,

that if μ = x + iy, y ̸= 0 is an eigenvalue of a real matrix

A with associated eigenvector v = u + iw, then the two real solutions

Y(t) = eat(u cos bt − wsin bt)

and

Z(t) = eat(u sin bt + wcos bt)

are linearly independent solutions of ˙X = AX

1b.Use (a) to solve the system

˙X =

(

3 1

−8 7

)

X.

NB: Real solutions are required.

Answer 1

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