1.2
n
Parameter Family of Solutions; General Solution; Particular Solu
tion
Introduction.
You know from your experience in previous mathematics courses that the calculus
of functions of several variables (limits, graphing, diﬀerentiation, integration and applications) is
more complicated than the calculus of functions of a single variable. By extension, therefore, you
would expect that the study of partial diﬀerential equations would be more complicated than the
study of ordinary diﬀerential equations. This is indeed the case! Since the intent of this material
is to introduce some of the basic theory and methods for diﬀerential equations, we shall conFne
ourselves to ordinary diﬀerential equations from this point forward. Hereafter the term
diﬀerential
equation
shall be interpreted to mean
ordinary diﬀerential equation.
Partial diﬀerential equations
are studied in subsequent courses.
±
We begin by considering the simple Frstorder diﬀerential equation
y
±
=
f
(
x
)
where
f
is some given function. In this case we can Fnd
y
simply by integrating:
y
=
±
f
(
x
)
dx
=
F
(
x
)+
C
where
F
is an antiderivative of
f
and
C
is an arbitrary constant. Not only did we Fnd
a
solution of the diﬀerential equation, we found a whole family of solutions each member of which is
determined by assigning a speciFc value to the constant
C
. In this context, the arbitrary constant
is called a
parameter
and the family of solutions is called a
oneparameter family
.
Remark
In calculus you learned that not only is each member of the family
y
=
F
(
x
C
a
solution of the diﬀerential equation but this family actually represents the set of
all
solutions of the
equation; that is, there are no other solutions outside of this family.
±
Example 1.
The diﬀerential equation
y
±
=3
x
2

sin 2
x
has the oneparameter family of solutions
y
(
x
)=
±
(
3
x
2

sin 2
x
)
dx
=
x
3
+
1
2
cos 2
x
+
C
As noted above, this family of solutions represents the set of all solutions of the equation.
±
In a similar manner, if we are given a second order equation of the form
y
±±
=
f
(
x
)
then we can Fnd
y
by integrating twice, with each integration step producing an arbitrary constant
of integration.
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 Spring '08
 Fonken
 Calculus, Derivative, Limits, Elementary algebra, Family of Solutions

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