A simply method for solving this problem is energy method. Initial extension of the spring in the mean position is

During oscillation, at any position x below the equilibrium position, let the velocity of m be v and angular velocity of the pulley be . If r is the radius of the pull, then .

At any instant, total energy for the system executing SHM is constant. Therefore

Substitute the value of

Taking derivative of both sides with respect to t,

The above equation is a second order differential equation of a system executing SHM.

Therefore,