# Homework Help: Homogenous ODE problem

1. Oct 20, 2012

### Baartzy89

Hi all,

I'm struggling with this question - I don't really know where to start. So far I have tried putting arbitrary values for 'a' into a quadratic auxiliary equation but using wolfram to calculate the roots gives me complex conjugates that I cant remember a thing about. Question as follows:

A body of mass 'm' kg attached to a spring moves with friction. The motion is described by the second Newton's law:

m(t).y" + a(t).y' + ky = 0

Where y is the body displacement in m, t is the time in s, a > 0 is the friction coefficient in kg/s and k is the spring constant in kg/s^2. Assuming m=1kg and k=4kg/s^2 find;

A) what is the range of values of a for which the body moves (i) with oscillations, (ii) without oscillation?

B) Find the general solution for any a<4 (Your solution should be a formula depending on the parameter a.) Proof that it follows from the solution obtained that the body slows down to a virtually rest state at large time (ie when t > infinity)?

C) find the particular solution for a=4 subject to the initial conditions y(0)=0, dy/dt= 1m/s at t=0. Plot this solution and determine the largest displacement of the mass usIng calculus.

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