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

    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 
   22 loadMacros(
   23 "PG.pl",
   24 "PGbasicmacros.pl",
   25 "PGchoicemacros.pl",
   26 "PGanswermacros.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() ;
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  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 ##################################################