Lecture Notes in Differential Equations - Bruce E. Shapiro

5
1.22 is restricted to being a positive number or zero. in the second case
(equation 1.25) there is no such restriction on C, and it is allowed to take
on any real value.
In the previous example we say that y = Ce t , where C is any arbitrary
constant is the general solution of the differential equation. A constant
like C that is allowed to take on multiple values in an equation is sometimes
called a parameter, and in this jargon we will sometimes say that y = Ce t
represents the one-parameter family of solutions (these are sometimes
also called the integral curves or solution curves) of the differential
equation, with parameter C. We will pin the value of the parameter down
more firmly in terms of initial value problems, which associate a specific
point, or initial condition, with a differential equation. We will return
to the concept of the one-parameter family of solutions in the next section,
where it provides us a geometric illustration of the concept of a differential
equation as a description of a dynamical system.
Definition 1.3 (Initial Value Problem (IVP)). An initial value problem
is given by
dy
= f(t, y)
dt
(1.26)
y(t 0 ) = y 0 (1.27)
where (t 0 , y 0 ) be a point in the domain of f(t, y). Equation 1.27 is called
an initial condition for the initial value problem.
Example 1.6. The following is an initial value problem:
⎫
dy
dt = 3t ⎬
⎭
y(0) = 27
(1.28)
Definition 1.4 (Solution, IVP). The function φ(t) is called a solution
of the initial value problem
⎫
dy
= f(t, y) ⎬
dt
⎭
y(t 0 ) = y 0
(1.29)
if φ(t) satisfies both the ODE and the IC, i.e., dφ/dt = f(t, φ(t)) and
φ(t 0 ) = y 0 .