Math 210, Spring 2013 Assignment #4, part B solutions

Math 210, Spring 2013
Assignment #4, part B solutions
O restart;
Q#1
Specify the differential equation and initial conditions
O de d diff u t , t = sin u t ;
de := d
u t = sin u t
dt
O
O
O
O
ic d u 0 = 1;
dsolve de, ic , u t ;
u t = arctan
ic := u 0 = 1
2 e t sin 1 K1 C cos 1
Ksin 1 2 K 2 e t 2 C 2 e t 2 cos 1 C e t 2 sin 1
K 2 et 2 cos 1 C sin 1 2 C e t 2 sin 1 2 t 2
K 2 e
Ksin 1 2 K 2 e t 2 C 2 e t 2 cos 1 C e t 2 sin 1 2
soln d rhs % ;
soln := arctan
2 e t sin 1 K1 C cos 1
Ksin 1 2 K 2 e t 2 C 2 e t 2 cos 1 C e t 2 sin 1
K 2 et 2 cos 1 C sin 1 2 C e t 2 sin 1 2 t 2
K 2 e
Ksin 1 2 K 2 e t 2 C 2 e t 2 cos 1 C e t 2 sin 1 2
plot soln, t =K10 ..10 ;
2 ,
2 ,
(1)
(2)
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Save the value of the exact solution at time 1 for comparison in the next question
O exact d eval soln, t = 1 ;
2 e sin 1 K1 C cos 1
exact := Karctan
2 e 2 cos 1 C sin 1 2 C e 2 sin 1 2 K 2 e 2
Q#2
O dsolve de, ic , u t , numeric ;
proc x_rkf45 ... end proc
O
O
O
O
% 1
% 2
rhs %
3
2
1
K10 K5 0 5
t
10
% K evalf exact
t = 1., u t = 1.95629507388886
u t = 1.95629507388886
1.95629507388886
1.03888862623691 10 -7
Let's set a small value for the options, "abserr" and "relerr" to make this error smaller.
C p
(5)
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I found I also needed to increase the number of digits used in floating point calculations to obtain the
desired accuracy
O Digits d 20;
Digits := 20
O
O
O
Q#3
Note that the output is supressed by default by putting a : after the "end do" statements, but that output is
forced by the "print" statement.
O for n from 2 to 6 do
u d 1;
N d 2 n ;
k d 1
N ;
eulit d u /evalf u C k$sin u ;
for i from 1 to N do
u d eulit u ;
end do:
errk n d evalf u K exact ;
print N, errk n ;
end do:
4, 0.015831482
O
numsol d dsolve de, ic , u t , numeric, abserr = 1eK12, relerr = 1eK12 ;
numsol := proc x_rkf45 ... end proc
rhs numsol 1 2 K evalf exact
Digits d 10;
8.052033 10 -13
Digits := 10
Following the steps from the posted lesson #6, with an added loop over the N value for the
total number of steps.
8, 0.006737062
16, 0.003078094
32, 0.001467045
64, 0.000715597
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