Stuck on these 2 hw problems would appreciate the help The p

Stuck on these 2 hw problems would appreciate the help

The phase plane equation for a linear oscillator can be used to directly obtain the solution, (a) Show that dt= dx/plusminus Squareroot 2E/m-k/m x^2 (b) Assume x = x_0 at t = 0. (Why is E Greaterthanorequalto (k/2)x_0^2 Integrate the above expression to obtain t as a function of x. Now solve for x as a function of t. Is your answer reasonable? Evaluate the following integral to determine the period T: T/4= integral_0^Squareroot 2E/K dx/Squareroot 2E/m-k/m x^2 Show that the period docs not depend on the amplitude of oscillation.

Solution

Suppose f is a periodic function with a period T = 2L. Then the Fourier series representation of f is a trigonometric series (that is, it is an infinite series consists of sine and cosine terms) of the form = = + + 1 0 cos sin 2 ( ) n n n L n x b L n x a a f x Where the coefficients are given by the Euler-Fourier formulas: = L L m dx L m x f x L a ( ) cos 1 , m = 0, 1, 2, 3, … = L L n dx L n x f x L b ( )sin 1 , n = 1, 2, 3, … The coefficients a’s are called the Fourier cosine coefficients (including a0, the constant term, which is in reality the 0-th cosine term), and b’s are called the Fourier sine coefficients.