I was working with the assumption:
Let r1, r2 = distinct eigenvalues of 2x2 matrix A where x' = Ax;
x = [y, v]
r1 = a+bi, where a, b are real
r2 = a-bi
x(t) = c1*e^(r1*t)*v1 + c2*e^(r2*t)*v2
=
I then jumped to writing
x(t) = e^(at)*(d1*cos(bt)*v1+d2*sin(bt)*v2).
But in reality, e*(x*i) = cos(x) + i*sin(x) so the proper formula is
x(t) = e^(at)*(c1*e^(bit)v1 + c2*e^(-bit)v2)
= e^(at)*(c1*(cos(bt)+i*sin(bt))v1 + c2*(cos(-bt)+sin(-bt))v2)
x(t) = e^(at)*( (c1*v1+c2*v2)*cos(bt) + i*(c1*v1-c2*v2)*sin(bt))
Such is the price of occasionally skipping the derivation.
In any event, the result
y(t) = e^(at)*( m1*cos(bt) + m2*sin(bt) )
v(t) = e^(at)*( n1*cos(bt) + n2*sin(bt) )
is unaffected.