Example 10
Symbolic and Numerical Solutions to Differential Equations I

Copyright © 1993 by Bill Titus, Carleton College, 
Department of Physics and Astronomy, Northfield, MN  55057-4025
September 6, 1993

Topics and Skills

Problem

Step 1 - The MMA command that symbolically solve differential equations (DEQ) is  DSolve[].  What can you find out about this command?

Comment 1

Step 2 - Activate the code below which solves the DEQ for  v[t], but without specifing any initial conditions.  Note that the derivative of  v[t]  is denoted by  v'[t] and that the output is called  soln1.

Clear[v, b, g, soln1]
soln1 = DSolve[v'[t] + b v[t] - g == 0, v[t], t]

Comment 2

Step 3 - Repeat Step 2 but add the initial condition  v[0] == vo  to the equation list.  Name the output  soln2.  Remember that there are now two equations and they should appear in a list.

Comment 3

Step 4 - See what happens if you replace  v[t]  by  v  in the second to last slot in the argument of  DSolve[].  Name this output  soln3.

Comment 4

Step 5 - Create a function  vel1[t, vo, b, g]  that corresponds to the velocity expression found in  soln2.  Show in two ways that the results agree with what you expect to happen as  b  goes to zero. (Hint:  what's  soln2[[1,1,2]] ?)

Step 6 - Create a plot, called  plot1,  of  vel1[t, vo, b, g]   for  0.0 < t < 30.0  when  b = 0.1,  g = 9.8,  and  vo = 0.0.  Be sure to label your axes.

Step 7 - Suppose the particle experiences a force due to air that's proportional to the cube of the particle's speed, i.e.,  Ff = -bv3.  Assuming  v[0]  is  vo, what's  v[t]?  Call the output to  DSolve[], in this case,  soln4.

Comment 7

Step 8 - What do you think is the MMA command for numerically solving DEQs?  Find out.

Comment 8

Step 9 - Numerically solve the DEQ for the cubic air resistance, assuming  b = 0.1,  g = 9.8,  and  vo = 0.0.  Consider the time interval from  t = 0.0  to  30.0.  Call the output   soln5.

Comment 9

Step 10 - Create the function   vel2[t]  that corresponds to the interpolating function in Step 9.  Then produce a plot, named  plot2,  of the velocity for  t = 0.0  to  t =  30.0.

Step 11 - Display both the linear and cubic results for the velocity of the particle on a single plot using  Show[].  Include the option

                   PlotRange -> All

Comment 11

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