08.05.1
Chapter 08.05
On Solving Higher Order Equations
for Ordinary Differential Equations
After reading this chapter, you should be able to:
1.
solve higher order and coupled differential equations
,
We have learned Euler’s and Runge-Kutta methods to solve first order ordinary differential
equations of the form
(
)
( )
0
0
,
,
y
y
y
x
f
dx
dy
=
=
(1)
What do we do to solve simultaneous (coupled) differential equations, or differential
equations that are higher than first order?
For example an
th
n
order differential equation of
the form
( )
x
f
y
a
dx
dy
a
dx
y
d
a
dx
y
d
a
o
n
n
n
n
n
n
=
+
+
+
+
−
−
−
1
1
1
1
(2)
with
1
−
n
initial conditions can be solved by assuming
1
z
y
=
(3.1)
2
1
z
dx
dz
dx
dy
=
=
(3.2)
3
2
2
2
z
dx
dz
dx
y
d
=
=
(3.3)
n
n
n
n
z
dx
dz
dx
y
d
=
=
−
−
−
1
1
1
(3.n)
( )
+
−
−
−
=
=
−
−
−
x
f
y
a
dx
dy
a
dx
y
d
a
a
dx
dz
dx
y
d
n
n
n
n
n
n
n
0
1
1
1
1
1
=
( )
(
)
x
f
z
a
z
a
z
a
a
n
n
n
+
−
−
−
−
1
0
2
1
1
1
(3.n+1)
The above Equations from (3.1) to (3.n+1) represent
n
first order differential equations as
follows

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