Graphics Programs Reference
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2. SolveProb.1with one step of the Runge-Kuttamethod of order (a)two and (b)
four.
3.
Integrate
y
=
sin
y
y
(0)
=
1
from
x
=
0to0
.
5with the second-order Taylor series methodusing
h
=
0
.
1. Com-
pare the result with Example 7.3.
4.
Verify that the problem
y
=
y
1
/
3
y
(0)
=
0
3)
3
/
2
.Which of the solutions wouldbe re-
producedbynumerical integrationif the initialconditionis set at(a)
y
hastwo solutions:
y
=
0 and
y
=
(2
x
/
=
0 and
10
−
16
?Verifyyour conclusions by integrating with any numerical method.
5. Convert the following differentialequations into first-order equations of the form
y
=
(b)
y
=
F
(
x
,
y
):
(a)ln
y
+
y
=
sin
x
y
y
xy
−
(b)
−
2
y
2
=
0
4
y
1
y
(4)
(c)
−
−
y
2
=
0
=
32
y
x
y
2
(d)
y
2
−
6.
In the following sets of coupleddifferentialequations
t
is the independent vari-
able.Convert these equations into first-order equations of the form
y
=
,
F
(
t
y
):
(a)
y
=
x
−
2
y
x
=
y
−
x
y
y
2
x
2
1
/
4
x
y
2
x
1
/
4
(b)
y
=−
+
x
=−
+
−
32
y
2
(c)
+
t
sin
y
=
4
x
x x
+
t
cos
y
=
4
y
7.
The differentialequation for the motion of a simple pendulum is
d
2
θ
dt
2
g
L
sin
=−
θ
where
θ
=
angular displacementfrom the vertical
g
=
gravitational acceleration
L
=
length of the pendulum
t
√
g
With the transformation
τ
=
/
L
the equationbecomes
d
2
θ
=−
sin
θ
d
τ
2
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