Question

Consider the following nonlinear differential equation, which models the unforced, undamped motion of a "soft" spring that does not obey Hooke's Law. (Here x denotes the position of a block attached to the spring, and the primes denote derivatives with respect to time t.) Note: x means x cubed notx a. Transform the second-order de. above into an equivalent system of first-order de.s. b. Use MATLAB's ode45 solver to generate a merical solution of this system aver the interval 0-t-6π for the following two sets initial conditions. x(0) = 2, z'(0) =-3 1. (0) 2, (0)0 c. Graph the two solutions on the same set of axes. Graph only r vs. t for each IVP; do not graph r. Be sure to label the axes and the curves. Include a title that contains your rame and describes the graph, something like "Numerical Solutions f r" +x-x3-0 by I. M. Smart." (obviously your name!). Make sure to include a date/time stamp on the graph, Note: To get " to appear in your title you will have to type command. in your MATLAB title d. Based on your graph, which solution appears to have the longer period? Explain clearly how you arrived at your answer

Answer 1

Plot:

MATLAB Scrlpt (run It as a scrpt, NOT from command window) close all clear clc 96 Letting u ti y1 - ode45(@vdpi, [O 6"pi], icl) t2 y2]- ode45(@vdpi [O 6pi], ic2 % Plotting figure, plot(t1. yl(L), t2, y2(:J) name 'NAME: 96 change this to your name xlabel('t'), ylabel(x(t)'), titlelsprintf("Numerical Solution to x""-x + x^3-0 by %s', name) legendr Conditions(O), Conditions ()) fprintf Based on the graph, () seems to have 4 periods in the given time span, while() seems to have 5 periods.In) fprinti(Hence, (i) has longer time period.In) function dydt vdp1f-y) dydt ly(2); y(1)-y(1)^3) 96 y(1-> x and y2)-> u or x' end