Graphics Programs Reference
In-Depth Information
weobtain
x
=
0
=
dv
dx
10
−
3
w
0
L
3
E I
19
.
444
×
x
=
L
=−
10
−
3
w
0
L
3
E I
dv
dx
22
.
222
×
10
−
3
w
0
L
4
E I
which agree with the analytical solution (easily obtained by direct integration of the
differentialequation).
v
|
x
=
0
.
5
L
=
6
.
5104
×
EXAMPLE 8.5
Solve the nonlinear differentialequation
4
x
y
3
y
(4)
+
=
0
with the boundary conditions
y
(0)
y
(1)
y
(1)
y
(0)
=
=
0
=
0
=
1
and plot
y
vs.
x
.
Solution
Our first task istohandle the indeterminacy of the differentialequation
at the origin, where
x
=
y
=
0. The problemis resolvedbyapplying L'Hospital's rule:
4
y
3
12
y
2
y
as
x
0. Thus the equivalent first-order equations and the boundary
conditionsthat we use in the solutionare
/
x
→
→
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
=
y
2
y
3
y
4
y
1
y
2
y
3
y
4
y
=
12
y
1
y
2
near
x
−
=
0
4
y
1
/
−
x
otherwise
y
1
(0)
=
y
2
(0)
=
0
y
3
(1)
=
0
y
4
(1)
=
1
Because the problemis nonlinear, we needreasonable estimates for
y
(0) and
y
(0). On the basis of the boundary conditions
y
(1)
0 and
y
(1)
=
=
1, the plot of
y
islikely to look something like this:
1
1
0
1
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