Lecture Notes in Differential Equations - Bruce E. Shapiro

14 LESSON 2. A GEOMETRIC VIEW
Example 2.2. Find and illustrate the one-parameter family of solutions
for the ODE
dy
dt = − t (2.6)
y
Cross multiplying and integrating
∫ ∫
ydy = −
tdt (2.7)
1
2 y2 = − 1 2 t2 + C (2.8)
Multiplying through by 2, bringing the t to the left hand side and redefining
C ′ = 2C gives us
y 2 + t 2 = C 2 (2.9)
which we (should!) recognize as the equation of a circle of radius C. The
curves for several values of C = 1, 2, . . . is illustrated in figure 2.3.
Figure 2.3: One parameter family of solutions for example 2.2.
6
5
4
3
2
1
0
1
2
3
4
5
6
6 5 4 3 2 1 0 1 2 3 4 5 6
Sometimes its not so easy to visualize the trajectories; a tool that gives us
some help here is the direction field. The direction field is a plot of the
slope of the trajectory. We know from the differential equation
dy
dt
= f(t, y) (2.10)