- 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.
- Find two linearly independent solutions to the differential equation. Use the Wronskian to prove that your solutions really are linearly independent.
- Find the solution that satisfies the initial conditions
*x*(0)=-2,*x*'(0) = 4.

- Consider the forced, undamped spring-mass differential equation shown
below.
- Find the general solution of the homogeneous differential equation.
- Find a particular solution.
- Find the general solution that satisfies the initial conditions
*x*(0)=1 and*x*'(0)=-2.

- Consider the following system of differential equations.
- Find the characteristic equation and determine its roots.
- Find the two linearly independent solutions. (You need not verify that they are linearly independent.)
- Find the solution that satisifies the initial condition
*x*(0)=0,*y*(0)=5. - Sketch the phase portrait for this system over the region , . Include at least four trajectories in your sketch.

- 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.