Consider the following nonlinear differential equation, whichmodels the unforced, undamped motion of a "soft" spring that doesnot obey Hooke's Law. (Here x denotes the position of a blockattached to the spring, and the primes denote derivatives withrespect to time t.) Note: x3 means x cubed not
x''' x?? - x + x^3 = 0
a. Transform the second-order d.e. above into an equivalentsystem of first-order d.e.’s.
b. Use MATLAB’s ode45 solver to generate a numerical solution ofthis system over the interval 0 ? t ? 6? for the following two setsof initial conditions.
i. x(0)=2,x?(0)=?3
ii. x(0) = 2, x?(0) = 0
c. Graph the two solutions on the same set of axes. Graph only xvs. t for each IVP; do not graph x?. Be sure to label the axes andthe curves. Include a title that contains your name and describesthe graph, something like “Numerical Solutions of x?? +x? x3 = 0 byI. M. Smart.” (obviously your name!). Make sure to include adate/time stamp on the graph, Note: To get x?? to appear in yourtitle you will have to type x???? in your MATLAB title command.
d. Based on your graph, which solution appears to have thelonger period? Explain clearly how you arrived at your answer
