Below are the graphs of 6 different solutions to mx" + cx' + kx = F₀cos(wt), x(0) = x₀, x'(0) = x₁ in red. For each graph you need to

1. Identify if the graph in red is
   • undamped?
   • underdamped?
   • overdamped?

   • unforced?
   • forced (beats)?
   • forced (resonance)?

2. Using what you determined in part (a), find the initial value problems for which the red graph is the graph of the solution.

Don't just randomly type in coefficients. Looking at the graphs of the solutions, you should be able to decide what the functions are (well, except for the transient part of one problem). Then think about what sort of problem has that solution. Note that this is actually closer to actual engineering practice than just solving equations. You have a behavior you want to get and you have to design the circuit to give you that behavior. While mathematically you have either positive or negative coefficients for an equation, in practical problems the coefficients are usually all positive since they have real meaning like mass or coefficient of friction, and those are positive. In the graphs below, the coefficients of the equations, though perhaps not the initial values, are all positive.

1.	x" +  x' +  x  =   cos(  t) x(0) =  x'(0) =
2.	x" +  x' +  x  =   cos(  t) x(0) =  x'(0) =
3.	x" +  x' +  x  =   cos(  t) x(0) =  x'(0) =
4.	x" +  x' +  x  =   cos(  t) x(0) =  x'(0) =
5.	x" +  x' +  x  =   cos(  t) x(0) =  x'(0) =
6.	x" +  x' +  x  =   cos(  t) x(0) =  x'(0) =
7.	In the above graphs you have examples of solutions that never cross the x-axis (problem 3) and that cross the x-axis infinitely many times (problems 1, 2, 5, and 6). It is hard to tell exactly how many times the solution crosses in problem 4 just from the graph. Is it possible to have a solution that crosses the axis exactly once? Give an example if you can or explain why it is impossible.
8.	Is it possible to have a solution that crosses the axis exactly twice? Give an example if you can or explain why it is impossible.