Graphics Programs Reference
In-Depth Information
Initial Value Problems
7
Solve
y
=
F
(
x
,
y
),
y
(
a
)
=
α
7.1
Introduction
The generalform of afirst-orderdifferentialequationis
y
=
f
(
x
,
y
)
(7.1a)
where
y
=
y
) is agiven function. The solution of thisequation contains
anarbitrary constant (the constantofintegration). To find thisconstant,wemust know
apointon the solution curve; that is,
y
must bespecifiedatsome valueof
x
,say at
x
dy
/
dx
and
f
(
x
,
=
a
.We write this auxiliary conditionas
y
(
a
)
=
α
(7.1b)
An ordinary differentialequation of order
n
f
x
y
(
n
−
1)
y
(
n
)
y
,...,
=
,
y
,
(7.2)
can always betransformedinto
n
first-order equations. Using the notation
y
y
y
(
n
−
1)
y
1
=
y
y
2
=
y
3
=
...
y
n
=
(7.3)
the equivalent first-order equations are
y
1
=
y
2
=
y
3
=
y
n
=
y
2
y
3
y
4
...
f
(
x
,
y
1
,
y
2
,...,
y
n
)
(7.4a)
The solutionnowrequires the knowledge
n
auxiliary conditions. If these conditions
arespecifiedat the same valueof
x
, the problemissaid to be an
initial value problem
.
Then the auxiliary conditions, called
initial conditions
,have the form
y
1
(
a
)
=
α
y
2
(
a
)
=
α
y
3
(
a
)
=
α
...
y
n
(
a
)
=
α
(7.4b)
1
2
3
n
251
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