The right-hand-side is not the scalar 0, but the vector 0 (constant function 0, in this case). Thus, c_1 cos(t) + c_2 sin(t) = 0 is equality of functions, not numbers. In other words, it holds for all values of t Substituting t = pi/4 gives c_1 + c_2 = 0 c_1 = -1, c_2 = 1 is one of its solutions, but so is c_1 = 0, c_2 = 0 Note that c_1 = 0, c_2 = 0 is the only solution that works for all values of t (as is shown in the video). Hence, c_1 = 0, c_2 = 0 is the only solution to the vector equation c_1 cos(t) + c_2 sin(t) = 0 Thoughts?