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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