Last Homework Problem of the Term 1 A vibrating spring can be modeled by a differential equation of the form x00 bx0 cx 0 where b and c are non negative constants The case b 0 corresponds to a frictionless system In this problem you ll consider the differential equations below in a couple of different ways making a connection between the second order equations and analyzing systems of differential equations i x00 14 x 0 ii x00 1 0 10 x 14 x 0 We ll begin with equation i corresponding to a frictionless system a Solve the second order differential equation Find the particular solution corresponding to the initial conditions x 0 1 and x0 0 0 Sketch the solution 2 b Equation i can be expressed ddt2x 14 x 0 Since x t gives position at time t v dx dt Then the differential equation can be written dv 1 x 0 dt 4 dx dt gives velocity Let Using both and gives the system of first order differential equations dx v dt dv 1 x dt 4 Analyze this system in the phase plane Check your work using the pplane applet c Using the system dx v dt and the relationship dv dx dv dt dx dt express dv dx dv 1 x dt 4 as a function of x and v The resulting differential equation is separable Solve Conclude that the trajectories in the phase plane are closed curves Does this make sense given a frictionless model d Which trajectory in the phase plane corresponds to the solution you drew in part a Now turn to equation ii x00 1 0 10 x 14 x 0 1 0 e Solve x00 10 x 14 x 0 Find the particular solution corresponding to the initial conditions x 0 1 and x0 0 0 Sketch the solution f Since x t gives position at time t Then the differential equation d2 x dt2 dx dt gives velocity Let v 1 dx 10 dt 1 4x dx dt 0 can be written dv 1 1 v x 0 dt 10 4 Using both and gives the system of first order differential equations dx v dt dv 1 1 v x dt 10 4 Analyze this system in the phase plane Check your work using the pplane applet 1 g Using the system dx v dt dv 1 1 v x dt 10 4 dv dv dv dt dx express dx as a function of x and v Is the resulting differential equation dx dt separable Solve only if it is separable and the relationship h Use the pplane applet to sketch a trajectory in the phase plane corresponding to the solution you drew in part f Do your pictures makes physical sense 2
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