Example 1.  Consider the D. E.  [Graphics:Images/ShootingMod_gr_58.gif]  over  [Graphics:Images/ShootingMod_gr_59.gif]  with  [Graphics:Images/ShootingMod_gr_60.gif]  and  [Graphics:Images/ShootingMod_gr_61.gif].
Is this a "linear differential equation" ?  Why ?
Identify the functions p(t), q(t) and r(t).  

Solution 1.

Look at the given D.E. and see how it is related to the form given in the theorem.

    [Graphics:../Images/ShootingMod_gr_62.gif]  with  [Graphics:../Images/ShootingMod_gr_63.gif]    

Our problem is seen to be

    [Graphics:../Images/ShootingMod_gr_64.gif] with  [Graphics:../Images/ShootingMod_gr_65.gif]  and  [Graphics:../Images/ShootingMod_gr_66.gif].  

Enter the function  p(t), q(t) and r(t) into Mathematica.  

 

 

[Graphics:../Images/ShootingMod_gr_67.gif]

Notice that q(t) is negative over the interval. Don't worry, the solution  [Graphics:../Images/ShootingMod_gr_68.gif]  can be computed with  [Graphics:../Images/ShootingMod_gr_69.gif].  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004