1550078481-Ordinary_Differential_Equations__Roberts_

396 Ordinary Differential Equations

Or

Thus, c 1 , c2, c3 and c 4 must simultaneously satisfy

C1 + C4/2 = 0

-C3 = 0

C2 + C3/2 = 0

C4 + 3/2 = 0.

Solving this system of equations, we find c 1 = 3/4, c2 = c3 = 0, and c4 =

• 3/2. Therefore, the solution of the initial value problem (13) is

Consequently,

Or, equivalently

u (t) = ~v 1 - ~[(sin2t)r +(cos 2t)s] + u p(t).

x(t)

x'(t)

y(t)

y' (t)

3 3

• 
  
  
  
  • 
    
    
    
    • cos 2t
      4 4
    
    
    
    
  
  
  
  

3. 2

2

sm t

-3 3t

-sin2t+ -

4 2

-3 3

-cos2t + -

2 2

Hence, the position of the electron in the xy-plane as a function of time, t, is

3 3 - 3 3t

(x(t), y(t)) = (

4

• 4
  cos2t,
  4

sin2t+

2

).

A graph of the path of the electron in the xy-plane is displayed in Figure 9. 7.
The electron is initially at the origin, travels along the path shown, and reaches
the point (0, l.5n) in 7r units of time.