Example 6.  Find the analytic solution to  [Graphics:Images/ShootingMod_gr_145.gif]  over  [Graphics:Images/ShootingMod_gr_146.gif]  with  [Graphics:Images/ShootingMod_gr_147.gif]  and  [Graphics:Images/ShootingMod_gr_148.gif].

Solution 6.

Now let Mathematica construct the analytic solution to the differential equation.  

[Graphics:../Images/ShootingMod_gr_149.gif]



[Graphics:../Images/ShootingMod_gr_150.gif]

[Graphics:../Images/ShootingMod_gr_151.gif]


Notice that this "real number" D. E. has a solution involving "complex numbers."
Ever wonder if we could "pull it off" ?

Onward to the boundary values.

[Graphics:../Images/ShootingMod_gr_152.gif]



[Graphics:../Images/ShootingMod_gr_153.gif]

[Graphics:../Images/ShootingMod_gr_154.gif]

[Graphics:../Images/ShootingMod_gr_155.gif]


Now dig out the formula and store it in the function z[t].

[Graphics:../Images/ShootingMod_gr_156.gif]



[Graphics:../Images/ShootingMod_gr_157.gif]

[Graphics:../Images/ShootingMod_gr_158.gif]


How could this ever be the solution ?  Dare we try to plot it ?

[Graphics:../Images/ShootingMod_gr_159.gif]


[Graphics:../Images/ShootingMod_gr_160.gif]

[Graphics:../Images/ShootingMod_gr_161.gif]

[Graphics:../Images/ShootingMod_gr_162.gif]

[Graphics:../Images/ShootingMod_gr_163.gif]

Now that wasn't so bad was it.

Hey!  Where's the numerical analysis.
If we wanted some sense out of that complex formula we can get it with "decimal" numbers.
First get rid of Mathematica's notation for the arbitrary constants, and simplify.
This takes Mathematica a very long time to think it out.  You must wait a minute while it thinks.

[Graphics:../Images/ShootingMod_gr_164.gif]



[Graphics:../Images/ShootingMod_gr_165.gif]

[Graphics:../Images/ShootingMod_gr_166.gif]
[Graphics:../Images/ShootingMod_gr_167.gif]


Numerically solve a system of equations for the constants [Graphics:../Images/ShootingMod_gr_168.gif]

[Graphics:../Images/ShootingMod_gr_169.gif]



[Graphics:../Images/ShootingMod_gr_170.gif]

[Graphics:../Images/ShootingMod_gr_171.gif]



[Graphics:../Images/ShootingMod_gr_172.gif]



[Graphics:../Images/ShootingMod_gr_173.gif]

[Graphics:../Images/ShootingMod_gr_174.gif]


Now form the function

[Graphics:../Images/ShootingMod_gr_175.gif]


[Graphics:../Images/ShootingMod_gr_176.gif]

Get rid of that complex number stuff.

[Graphics:../Images/ShootingMod_gr_177.gif]


[Graphics:../Images/ShootingMod_gr_178.gif]

Looks like a good old fashioned function composed of "well known" functions.

[Graphics:../Images/ShootingMod_gr_179.gif]



[Graphics:../Images/ShootingMod_gr_180.gif]

[Graphics:../Images/ShootingMod_gr_181.gif]

[Graphics:../Images/ShootingMod_gr_182.gif]
[Graphics:../Images/ShootingMod_gr_183.gif]

[Graphics:../Images/ShootingMod_gr_184.gif]
[Graphics:../Images/ShootingMod_gr_185.gif]
[Graphics:../Images/ShootingMod_gr_186.gif]

[Graphics:../Images/ShootingMod_gr_187.gif]
[Graphics:../Images/ShootingMod_gr_188.gif]
[Graphics:../Images/ShootingMod_gr_189.gif]
[Graphics:../Images/ShootingMod_gr_190.gif]
[Graphics:../Images/ShootingMod_gr_191.gif]

[Graphics:../Images/ShootingMod_gr_192.gif]


Now plot the analytic solution to the differential equation.  

[Graphics:../Images/ShootingMod_gr_193.gif]



[Graphics:../Images/ShootingMod_gr_194.gif]

[Graphics:../Images/ShootingMod_gr_195.gif]

[Graphics:../Images/ShootingMod_gr_196.gif]

[Graphics:../Images/ShootingMod_gr_197.gif]

[Graphics:../Images/ShootingMod_gr_198.gif]

[Graphics:../Images/ShootingMod_gr_199.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004