View of /trunk/NationalProblemLibrary/Rochester/setDiffEQ13Systems1stOrder/ur_de_13_10.pg

    1 ## DESCRIPTION
    2 ## Calculus
    3 ## ENDDESCRIPTION
    4 
    5 ## KEYWORDS('differential equation' 'first order' 'matrices')
    6 ## Tagged by tda2d
    7 
    8 ## DBsubject('Calculus')
    9 ## DBchapter('Differential Equations')
   10 ## DBsection('Systems of Differential Equations')
   11 ## Date('')
   12 ## Author('')
   13 ## Institution('Rochester')
   14 ## TitleText1('')
   15 ## EditionText1('')
   16 ## AuthorText1('')
   17 ## Section1('')
   18 ## Problem1('')
   19 
   20 
   21 DOCUMENT();        # This should be the first executable line in the problem.
   22 
   23 loadMacros(
   24 "PG.pl",
   25 "PGbasicmacros.pl",
   26 "PGchoicemacros.pl",
   27 "PGanswermacros.pl",
   28 "PGgraphmacros.pl",
   29 "PGmatrixmacros.pl",
   30 "PGnumericalmacros.pl",
   31 "PGauxiliaryFunctions.pl",
   32 "PGmorematrixmacros.pl"
   33 );
   34 
   35 TEXT(beginproblem());
   36 $showPartialCorrectAnswers = 1;
   37 
   38 $v11 = random(1,3,2);
   39 $v21 = random(1,2,1);
   40 $v12 = 3 - $v21;
   41 $v22 = 4 - $v11;
   42 
   43 $S = new Matrix(2,2);
   44 $S -> assign(1,1,$v11);
   45 $S -> assign(1,2,$v12);
   46 $S -> assign(2,1,$v21);
   47 $S -> assign(2,2,$v22);
   48 
   49 $Sinv = new Matrix(2,2);
   50 $Sinv -> assign(1,1, $v22);
   51 $Sinv -> assign(1,2, - $v12);
   52 $Sinv -> assign(2,1, - $v21);
   53 $Sinv -> assign(2,2, $v11);
   54 
   55 $eig1 = random(2,5,1) * random(-1,1,2);
   56 $eig2 = random(2,5,1) * random(-1,1,2);
   57 if ($eig2 == $eig1) { $eig2 = - $eig2; }
   58 
   59 $e = new Matrix(2,2);
   60 $e -> assign(1,1,$eig1);
   61 $e -> assign(1,2,0);
   62 $e -> assign(2,1,0);
   63 $e -> assign(2,2,$eig2);
   64 
   65 $c1 = non_zero_random(-5,5,1);
   66 $c2 = non_zero_random(-5,5,1);
   67 
   68 $M = new Matrix(2,2);
   69 $M = $S * $e * $Sinv;
   70 
   71 $rhs = new Matrix(2,1);
   72 $rhs -> assign(1,1, $c1*$v11 + $c2*$v12);
   73 $rhs -> assign(2,1, $c1*$v21 + $c2*$v22);
   74 
   75 
   76 BEGIN_TEXT
   77 
   78 Solve the system
   79 $BR
   80 \{ mbox( "\( \displaystyle \frac{dx}{dt} = \)", display_matrix($M), "\(x\)" ) \}
   81 $BR
   82 with the initial value \{ mbox( "\( x(0) = \)", display_matrix($rhs), "." ) \}
   83 $BR
   84 \{ mbox( "\( x(t) = \)", answer_matrix(2,1,40), "." ) \}
   85 
   86 END_TEXT
   87 
   88 $ans1 = "$c1*e^($eig1*t)*$v11 + $c2*e^($eig2*t)*$v12";
   89 $ans2 = "$c1*e^($eig1*t)*$v21 + $c2*e^($eig2*t)*$v22";
   90 
   91 ANS(fun_cmp($ans1, vars => 't'));
   92 ANS(fun_cmp($ans2, vars => 't'));
   93 
   94 ENDDOCUMENT();       # This should be the last executable line in the problem.
   95