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
- cos 2t
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.