1. HARMONIC AND ANHARMONIC OSCILLATORS This is a modified version of chapter 3 problem 4 in your textbook. For simple harmonic motion, the equation of motion is given by d²x dt² =-kxa with a = 1. Write a program that uses the Euler-Cromer method to solve for x(t). For convenience, let a = k = 1. Show that the period of the oscillations is independent of the amplitude of the motion (this is a key feature of simple harmonic motion!). Use several different regularly-spaced values of amplitude and show that the amplitude and period are independent of each other. Write a program that uses the Euler-Cromer method to solve for x(t) when k = 1 but a = 3 and plot a representative solution for x(t). Calculate the period of the oscillation for a variety of amplitudes. Show that the period of oscillation is now dependent on the amplitude of the motion (this is called anharmonic motion). Describe the anharmonic behavior that you observe. Provide an intuitive, qualitative discussion of why this behavior is observed.