Graphics Programs Reference
In-Depth Information
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
Search WWH ::

Custom Search