Lecture Notes in Differential Equations - Bruce E. Shapiro

13
We can extend this geometric interpretation from the initial value problem
to the general solution of the differential equation. For example we found in
example 1.5 that y = Ce t is the solution to the ODE y ′ = y, and we called
the expression the one-parameter family of solutions. To see what this
means consider the effect of the initial condition on y = Ce t : it determines
a specific value for C. In fact, if we were to plot every conceivable curve
y = Ce t (for every value of C), the picture would look something like those
shown in figure 2.2 The large black dot indicates the location of the point
Figure 2.2: Illustration of the one parameter family of solutions found in
example 1.5.
y(t)
3
2
1
0
1
2
C=1.5
C=1.0
C=.5
C=0
C= -5
3
1. 0.5 0. 0.5 1.
t
(0, 1), and the values of C are shown for several of the curves. 4 We see that
the curve corresponding to C = 1.0 is the only curve that passes through
the point (0, 1) - this is a result of the uniqueness of the solutions. As long
the conditions of the fundamental theorem (theorem 1.5) are met, there
is always precisely one curve that passes through any given point. The
family (or collection) of curves that we see in this picture represents the
one-parameter family of solutions: each member of the family is a different
curve, and is differentiated by its relatives by the value of C, which we call
a parameter. Another term that is sometimes used for the one-parameter
family of solutions is the set of integral curves.
4 In fact, not all curves are shown here, only the curves for C =
. . . , −.5, 0, 0.5, 1, 1.5, . . . . Curves for other values fall between these.