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### Motion Lab

You've done a lot of problems so far where we have an equation and you
decide what the solution is. In this lab we are going to reverse the
process. Below are the graphs of 7 different solutions to equations of the
form $mx''+cx'+kx=F_0 \cos(\omega t)$, $x(0)=x_0$, $x'(0)=x_1$. You are to
find the initial value problems for which they are solutions. To help,
we've got an applet that will let you input an initial value problem and
graph the solution. However, 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 have to design the
circuit to give it to you. While mathematically you can 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 or friction and those are positive). In the
graphs below, the coefficients of the equation, *though perhaps not
the initial values*, are all positive.

If you have any problems with this page, please contact bennett@math.ksu.edu.

©2010 Andrew G. Bennett

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Launch the motion lab.
Change coefficients as you want, then click the "Update Graph" button to
see the graph of the solution. The "Reset" button will return the
coefficients to the last set of values that you graphed. This means that
while you are adjusting coefficients the equation and the graph will
temporarily be out of sync, but pressing either button will bring them
back into sync, either by making the graph match the equation (Update
Graph) or resetting the equation to match the graph (Reset). ** Note that
some problems have different graph windows and you should update the graph
window on the applet to agree with the picture you are trying to
match.**

- Can you find an
**unforced problem**where the solution curve never crosses the axis?

Can you find an unforced problem where the solution curve crosses the axis exactly once?

Can you find an unforced problem where the solution curve crosses the axis exactly twice?

Can you find an unforced problem where the solution curve crosses the axis infinitely many times?

If you have any problems with this page, please contact bennett@math.ksu.edu.

©2010 Andrew G. Bennett