A method of solving Mathieu's equation

MATHIEU'S equation ${d^{2}y \over dt^{2}} + (b - 2q \cos 2t)y = 0 \eqno{\hbox{(1)}}$ is gradually becoming a very important equation in physics and engineering. It apparently first appeared in a paper by Mathieu1 on the vibration of an elliptic membrane. However, since that time it has made its appearance in many practical problems in the fields of mechanics, electricity and magnetism, sound, etc. Many such examples will be found in McLachlan's book on Mathieu functions.2