For the set of differential equations (dv/dt)=3v-13w (dw/dt)=5v+1w with initial

Question

For the set of differential equations
(dv/dt)=3v-13w
(dw/dt)=5v+1w

with initial conditions Vo= 3 and Wo= -10 at t= 0.
(a) Rewrite the above set of equations in the form (A-lemdaI)x=0; what is A?
(b) What are the eigenvalues Lemda?
(c) What is the trace? Show that the sum of the eigenvalues is equal to the trace.
(d) What are the associated eigenvectors?
(e) Use the initial conditions to get the full solutions for v(t) and w(t)? Make sure to
expand any complex exponentials into sines and cosines.

Explanation / Answer

a) 3 -13 5 1 b) 2+8i and 2-8i c) 4 and sum of lamda = 2+8i + 2-8i = 4 (equal) d) 0.2 -1.6i

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