Introduction 171.3 Solutions and Problems
In the previous section , we learned what a solution to a differential equ a-
tion is and how to ch eck if a p articular function is a solution to a specified
differential equation or not. At this point, you might ask: "How many solu-
tions does a differential equation have?" The short answer is "It depends on
the differential equation." For instance, the differential equation IY'I + 1 = 0
has no solution, while the differential equation (y')^2 + y^2 = 0 has exactly onesolut ion- t he function y(x ) = 0. By integration, we find that the solution of
t he differentia l equation y' = 2x is
(1) y(x) = x^2 + C
where C is an arb itrary constant. Hence, the differential equation y' = 2x has
a n infinite number of solutions- one solution for each choice of the value of
the constant C. The set of solut ions y(x) = x^2 +C is called a one-parameter
family of solutions of t he differential equation y' = 2x. The graph of this
family of solut ions is called the integral curves or solution curves of the
differential equation. A solution of a differential equation which contains
no arbitrary constants is called a particular solution. Choosing C = 1 in
equation (1), we obtain the particular solut ion y(x) = x^2 +1 of the differential
equation y' = 2x. A graph of the particular solutions obtained from (1) by
choosing C = -2, C = -1, C = 0, C = 1, and C = 2 are shown in Figure 1.2.
The graphs of these solution curves are parabolas which open upward, h ave
vertices at (0 , C), and have the y-axis as their axes of symmetry. Thus,
Figure 1.2 represents the solution curves of the differential equation y' = 2x.
x-4^4
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Figure 1.2 Solution Curves of the Differential Equation y' = 2x