Monday, November 17, 2008

Differential Equations Project Tips

As some questions are asked more regularly, I thought I'd provide some general discussion here.

(1) Start with the proposed form of X(t).  Compute X'(t) and X''(t) based on that form.  Then use those calculations to discover when X'' + k/m X = 0.

(2) The question "mean physically about the mass on a spring" is not asking you to think about the mass (as in measurement) but is asking you to think about what the statement X(0)=1 means about the state of the mass at time t=0 and what the statement X'(0)=0 means about the state of the mass at time t=0.

(3) X(0) is a constant and has derivative of d/dt[X(0)] = 0.  Recall that dX/dt > 0 implies that X is increasing, dX/dt < 0 implies that X is stationary (instantaneous rate = 0)

(4) An arbitrary quantity A is proportional to some other quantity B if it is always that case that A = k B for some constant value k (the constant of proportionality).  Now interpret the statements to identify what pieces of the equation are proportional to what.

(5) Although you have studied ex in precalculus, you will not use the logarithm at all in this work.  Instead, I just want you to consider some function that has the special property that exp' = exp.  (This sentence is analogous to sin'=cos and cos'=-sin.)  However, you do need to think about the chain rule: X(t) = A exp(rt)  (Since the argument is not simply t, you must use the chain rule.)  This problem is exactly analogous to Step 1.

(6) You will get something like
dX/dt = "formula involving X and a, b, and m"
X is increasing when "formula" > 0, decreasing when "formula" < 0, and stationary when "formula" = 0.  So use your skills with algebra (think sign analysis) to find conditions when these are the case.

(7) You need to understand the relationship between a rate of change and an actual change.  To understand this as well as possible, see the section we skipped in Chapter 3 (last section).  But what you essentially need is that we will follow the tangent line for the time increment Δt.  How much change is there when the rate of change and the duration of time are both known?

(8) The new version of Excel has some unanticipated differences from what I had when I wrote the project.  The labels are not assigned from a menu anymore.  Instead of the 3-step process that is described, you just click in the label field in the header section of Excel and type in the new label and then hit enter.

The calculations you see in the first few lines should exactly match your hand calculations in part (7).  

Do not print the spreadsheet (it takes WAY too many pages).  That is why I ask you to submit your spreadsheet on Blackboard as part of the project.

(9) I must receive a print out of the graph -- hand drawn figures are not acceptable.   Ideally, this entire project report would be typed (perhaps using Equation Editor for the equations), with the figures naturally fitting in.

(10) Make hypotheses and test your hypotheses.

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