Consider the nonlinear springmass equation m d2xdt2 x2 Deri

Consider the non-linear spring-mass equation m d^2x/dt^2 = -x^2. Derive a conservation equation. On a single plot, graph several different types of representative trajectories in the phase plane (dx/dt vs. x) for m = 1. Include arrows. Include trajectories that \"escape to negative infinity\" as well as trajectories that tend towards the equilibrium position. For the trajectory that tends towards the equilibrium position, examine how long it takes to reach the equilibrium position by doing the following: Set the conserved quantity in (a.) to 0 since dx/dt = x = 0 at the equilibrium position. Solve the differential equation you obtained in (i.), recognizing that on this trajectory we have x 0. From your result in (ii.), determine what is true about t as x rightarrow 0.

Solution

md^x/dt^2= -x^2

integrating

mdx/dt=-x^3/3

b. i. if we set this equal to zero then dx/dt=x

dx/x=dt

so, integrating

ln x = t +c

or, x= e^ct

as x approaches zero t will approach infinity cinsidering c being negative