- Consider the following differential equation, describing a
damped spring-mass system.
- Find the roots of the characteristic equation. Classify
the spring-mass system
as underdamped, overdamped, or critically damped.
The characteristic equation is

The roots are complex, .

- Find two linearly independent solutions to the
differential equation. Use the Wronskian to prove that your solutions
really are linearly independent.
The two solutions are
and

The Wronskian is given by the determinant

After simplification, and using the trig identity , we get

- Find the solution that satisfies the initial conditions
*x*(0)=-2,*x*'(0) = 4. The general solution isand the derivative is

So plugging in

*t*=0 gives the two equationsSolving gives , .

- Find the roots of the characteristic equation. Classify
the spring-mass system
as underdamped, overdamped, or critically damped.
- Consider the forced, undamped spring-mass differential equation shown
below.
- Find the general solution of the homogeneous differential
equation.
The general solution to the homogeneous equation is

- Find a particular solution.
The forcing term is a solution to the homogeneous equation, so the
particular solution must have the form
This gives

so

Comparing to the forcing term, we get

*A*=1,*B*=0, so - Find the general solution that satisfies the initial conditions
*x*(0)=1 and*x*'(0)=-2.The general solution is

The derivative is

so the equations to be satisfied are

- Find the general solution of the homogeneous differential
equation.
- Consider the following system of differential equations.
- Find the characteristic equation and determine its roots.
The characteristic equation is

So the roots are

*r*=-1 and*r*=-3. - Find the two linearly independent solutions. (You need
not verify that they are linearly independent.)
For

*r*=-3, the equation is*A*+*B*=0 so a good choice is*A*=1,*B*=-1. For*r*=-1, the equation is -*A*+*B*=0 so a good choice is*A*=1,*B*=1. So the two linearly independent solutions areand

- Find the solution that satisifies the initial condition
*x*(0)=0,*y*(0)=5.The equations to be satsified are

so and . The solution is

- Sketch the phase portrait for this system over the region , . Include at least four trajectories in
your sketch.

- Find the characteristic equation and determine its roots.
- On the direction field given below, sketch trajectories starting
at the initial conditions
*x*(0)=-1,*y*(0)=0.5 and*x*=1.5,*y*(0)=0.

*William W. Farr*

Mon May 5 08:48:38 EDT 1997