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.