[npl] / trunk / NationalProblemLibrary / Rochester / setDiffEQ9Linear2ndOrderHomog / ur_de_9_10.pg Repository: Repository Listing bbplugincoursesdistsnplrochestersystemwww

# View of /trunk/NationalProblemLibrary/Rochester/setDiffEQ9Linear2ndOrderHomog/ur_de_9_10.pg

Fri Apr 28 13:46:21 2006 UTC (7 years ago) by jjholt
File size: 3050 byte(s)
Added tags.  --JH


    1 ## DESCRIPTION
2 ## Calculus
3 ## ENDDESCRIPTION
4
5 ## KEYWORDS('differential equation' 'second order' 'linear')
6 ## Tagged by tda2d
7
8 ## DBsubject('Calculus')
9 ## DBchapter('Second-Order Differential Equations')
10 ## DBsection('Second-Order Linear Equations')
11 ## Date('')
12 ## Author('')
13 ## Institution('Rochester')
14 ## TitleText1('')
15 ## EditionText1('')
16 ## AuthorText1('')
17 ## Section1('')
18 ## Problem1('')
19
20 DOCUMENT() ;
21
23 "PG.pl",
24 "PGbasicmacros.pl",
25 "PGchoicemacros.pl",
27 "PGauxiliaryFunctions.pl",
28 "PGdiffeqmacros.pl"
29 ) ;
30 ################
31 $aa = random(2,11,1) ; 32$a = $aa **2 ; 33$B = random(-10,10,1) ;
34 $e = random(1,9,1) ; 35$b = 2*$B*$aa ;
36 $c =$B *$B - ($e **2) ;
37 $m = random(1,9,1) ; 38$n = random(1,9,1) ;
39 ###################
40
41
42 $L = diffop($a,$b,$c,$m,$n) ;
43 $y_1 = ivy($a,$b,$c,1,0) ;
44 $y_2 = ivy($a,$b,$c,0,1) ;
45 @abe = frac(-$b,$a) ;
46 $abel =$abe[0] ;
47 $W = "exp(-$b/$a *t)" ; 48 49 TEXT(beginproblem()) ; 50 51$showPartialCorrectAnswers = 1 ;
52
53 BEGIN_TEXT
54
55 Find  the function   $$y_1$$  of $$t$$  which is the solution of
56 $L = 0$
57 with initial conditions $$\quad y_1(0) = 1, \quad y_1'(0) = 0 .$$ $BR 58 $$y_1 =$$ \{ans_rule(60)\}$BR $BR 59 Find the function $$y_2$$ of $$t$$ which is the solution of 60 $L = 0$$BR
61  with initial conditions $$\quad y_2(0) = 0, \quad y_2'(0) = 1 .$$ $BR 62 $$y_2 =$$ \{ans_rule(60)\}$BR $BR 63 Find the Wronskian $W(t) = W(y_1,y_2).$$BR
64 $$W(t) =$$ \{ans_rule(60)\} $BR$BR
65 Remark: You can find W by direct computation and use
66 Abel's theorem as a check.  You should find that W is not zero
67 and so $$y_1$$ and $$y_2$$ form a fundamental set of solutions of
68 $L = 0.$   $BR 69 END_TEXT 70 71 ANS(fun_cmp($y_1, vars=>"t")) ;
72 ANS(fun_cmp($y_2, vars=>"t")) ; 73 ANS(fun_cmp($W, vars=>"t")) ;
74
75 ENDDOCUMENT() ;
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102 ##################################################
103 my $XML_INFORMATION = <<'END_OF_XML_TRAILER_INFO'; 104 <?xml version="1.0"?> 105 <metaPGdata> 106 <author>David Prill</author> 107 <course>MTH163</course> 108 <description>Differential equations 109 ay'' + by' + cy = 0 110$a,$b,$c should be integers and $a>0.</description> 111 <fullPath>setDESOLinear/11.pg</fullPath> 112 <institution>University of Rochester</institution> 113 <keywords>Differential Equation,Initial value problem, 114 second order linear,constant coefficients,rational roots</keywords> 115 <libraryPath>setDESOLinear/11.pg</libraryPath> 116 <libraryURL>http://webhost.math.rochester.edu/mth163lib/discuss/msgReader$399</libraryURL>
117         <modified><dateTime.iso8601>20000718T12:56:25</dateTime.iso8601></modified>
118         <msgNum>399</msgNum>
119         <pgProblem>true</pgProblem>
120         <preface></preface>
121         <problemVariants></problemVariants>
122         <probNum></probNum>
123         <psvn></psvn>
124         <revisedVersions></revisedVersions>
125         <setName>DESOLinear</setName>
126         <titleRoot>11</titleRoot>
127         </metaPGdata>
128
129 END_OF_XML_TRAILER_INFO
130 ##################################################