View of /trunk/NationalProblemLibrary/FortLewis/DiffEq/2-Higher-order/02-Linear-2nd-order-cc/Lebl-2-2-11.pg

    1 ## DESCRIPTION
    2 ## Higher order ODEs: constant coefficient second order homogeneous linear ODEs
    3 ## ENDDESCRIPTION
    4 
    5 ## KEYWORDS('differential equations','second order linear ODE','constant coefficients')
    6 
    7 ## DBsubject('Differential Equations')
    8 ## DBchapter('Higher Order Differential Equations')
    9 ## DBsection('Second Order Linear')
   10 ## Date('01/30/2011')
   11 ## Author('Paul Pearson')
   12 ## Institution('Fort Lewis College')
   13 ## TitleText1('Notes on Diffy Qs')
   14 ## EditionText1('December 9, 2010')
   15 ## AuthorText1('Jiri Lebl')
   16 ## Section1('2.2')
   17 ## Problem1('11')
   18 
   19 
   20 ##############################
   21 #  Initialization
   22 
   23 DOCUMENT();
   24 
   25 loadMacros(
   26 "PGstandard.pl",
   27 "MathObjects.pl",
   28 "AnswerFormatHelp.pl",
   29 "parserAssignment.pl",
   30 );
   31 
   32 TEXT(beginproblem());
   33 
   34 
   35 #############################
   36 #  Setup
   37 
   38 Context("Numeric");
   39 Context()->variables->add(
   40 A=>"Real",B=>"Real",y=>"Real",
   41 );
   42 Context()->flags->set(
   43     formatStudentAnswer=>'parsed',
   44     reduceConstants=>0,
   45     reduceConstantFunctions=>0,
   46 );
   47 parser::Assignment->Allow;
   48 
   49 $a  = random(2,5,1);
   50 $aa = $a**2;
   51 $a2 = 2*$a;
   52 do { $b = random(2,5,1); } until ($b != $a);
   53 $bb = $b**2;
   54 $c = $aa + $bb;
   55 
   56 $s = random(-1,1,2);
   57 $msa = -($s) * $a;
   58 
   59 # char poly (r + $s $a)^2 + $b^2
   60 
   61 if ($s==1) {
   62   $diffeq = "y'' + $a2 y' + $c y = 0"; # tex
   63 } else {
   64   $diffeq = "y'' - $a2 y' + $c y = 0"; # tex
   65 }
   66 
   67 
   68 $answer = Compute("y = A e^($msa x) cos($b x) + B e^($msa x) sin($b x)");
   69 
   70 
   71 ######################
   72 #  Main text
   73 
   74 Context()->texStrings;
   75 BEGIN_TEXT
   76 Find the general solution to \( $diffeq \).
   77 In your answer, use \( A \) and \( B \) to
   78 denote arbitrary constants and \( x \)
   79 the independent variable.
   80 $BR
   81 $BR
   82 \{ ans_rule(40) \}
   83 \{ AnswerFormatHelp("equations") \}
   84 END_TEXT
   85 Context()->normalStrings;
   86 
   87 
   88 ######################
   89 #  Answer evaluation
   90 
   91 $showPartialCorrectAnswers = 1;
   92 ANS( $answer->cmp( checker => sub {
   93     my ( $correct, $student, $self ) = @_;
   94     my $stu   = Formula($student->{tree}{rop});
   95 
   96     ################################
   97     #  Check for arbitrary constants
   98     #
   99     Value->Error("Is your answer the most general solution?")
  100     if (
  101       Formula($stu->D('A'))==Formula(0) ||
  102       Formula($stu->D('B'))==Formula(0)
  103     );
  104 
  105     ############################################
  106     #  Check for linear independence (Wronskian)
  107     #
  108     my $x = Real(1.24);
  109 
  110     my $a11 = $stu->eval(A=>1,B=>0,x=>$x,y=>0);
  111     my $a12 = $stu->eval(A=>0,B=>1,x=>$x,y=>0);
  112 
  113     my $a21 = $stu->D('x')->eval(A=>1,B=>0,x=>$x,y=>0);
  114     my $a22 = $stu->D('x')->eval(A=>0,B=>1,x=>$x,y=>0);
  115 
  116     my $wronskian = $a11*$a22 - $a21*$a12;
  117 
  118     Value->Error("Your functions are not linearly independent or your answer is not complete")
  119     if ($wronskian==Real(0));
  120 
  121     ########################################################
  122     #  Check that the student answer is a solution to the DE
  123     #
  124     my $stux  = Formula($stu->D('x'));
  125     my $stuxx = Formula($stu->D('x','x'));
  126     my $stuDE = Formula($stuxx + $s*$a2*$stux + $c*$stu)
  127     ->with(test_points=>[[1,1,0.1,0],[2,1,0,0],[1,2,-0.1,0],[1,1,0,1]]);
  128 
  129     return ($stuDE==Formula(0));
  130 
  131 }));
  132 
  133 
  134 COMMENT("MathObject version.");
  135 
  136 ENDDOCUMENT();