3x3-system-convert.pg

## DESCRIPTION
## Differential Equations.  
## ENDDESCRIPTION

## DBsubject(Differential equations)
## DBchapter(Systems of differential equations)
## DBsection(Reduction to first order systems)
## Institution(METU-NCC)
## KEYWORDS('differential equation' 'first order' 'matrices')


DOCUMENT();
loadMacros(
  "PGstandard.pl",
  "PGmatrixmacros.pl",
  "PGbasicmacros.pl",    # needed for answer_matrix
  "MathObjects.pl",
  "parserMultiAnswer.pl",
  "answerHints.pl",
);

Context("Numeric");
Context()->variables->{namePattern} = qr/[a-z][a-z0-9_]*'*/i;
Context()->variables->are(
     t=>"Real",
     y=>"Real",
  "y'"=>"Real",
 "y''"=>"Real",
"y'''"=>"Real",
  "x1"=>"Real",
  "x2"=>"Real",
  "x3"=>"Real"
);

sub random_F {
  my $myF; 
  my $c = non_zero_random(-3,3,1);

  if (random(0,2,1)) {
    my $max_n = random(1,4,1);
    my $terms = random(1,min($max_n,3),1);
    $myF = "$c t^$max_n";
    my @all_n = (0 .. ($max_n-1)); 
    for (my $i=1; $i<$terms; $i++) {
      $c = non_zero_random(-3,3,1);
      my $n = splice(@all_n,random(0,@all_n-1,1),1);
      $myF = $myF . " + $c t^$n";
    }
  } else {
    if (random(0,1,1)) {
      $myF = "$c cos(t)";
    } else {
      $myF = "$c sin(t)";
    } 
  }
  return Formula("$myF")->reduce;
}

my $p0, $p1, $p2, $p3, $g;

$p0=non_zero_random(-4,4,1); 

# $p0 = random_F(); 
$p2 = random_F(); 
 $g = random_F();

if (random(0,1,1)) { $p1=random_F(); } 
  else { $p1 = Formula("0"); }
if (random(0,1,1)) { $p3=random_F(); }
  else { $p3 = Formula("0"); }

$t0 = random(-3,3,1);
$y0 = random(-3,3,1); $yp0 = random(-3,3,1); $ypp0 = random(-3,3,1);


$Diff_Eq = Formula("$p0 y''' + $p1 y'' + $p2 y' + $p3 y ")->reduce;

Context()->noreduce('(-x)-y','(-x)+y');
$Diff_Eq_r1 = Formula(" - ($p3 / $p0) y - ($p2 / $p0) y' - ($p1 / $p0) y'' + ($g / $p0)")->reduce;
$Diff_Eq_r2 = Formula(" - ($p3 / $p0) x1 - ($p2 / $p0) x2 - ($p1 / $p0) x3 + ($g / $p0)")->reduce;

Context()->texStrings;
BEGIN_TEXT

Write the given third order linear equation as an equivalent system 
of first order equations with initial values.
\[ $Diff_Eq = $g \\ 
   {\ }\qquad \qquad \mathrm{with}\qquad 
   y($t0)=$y0,\ \ y'($t0)=$yp0,\ \ y''($t0)=$ypp0 \] 
Use \(x_1 = y\), \(x_2 = y'\), and \(x_3 = y''\).
$PAR
\{ mbox( 
 "\( \displaystyle \boldsymbol{\vec{x}\,'} = \)", 
 answer_matrix(3,3,10), 
 "\( \displaystyle \boldsymbol{\vec{x}} \) + ",
 answer_matrix(3,1,10) 
) \}  
$BR
with initial values
\{ mbox(
 "\(\qquad \boldsymbol{\vec{x}} \biggl( \)", 
 ans_rule(5),
 "\( \biggr) = \)",
 answer_matrix(3,1,5)
) \}

END_TEXT

$showHint = 2;

BEGIN_TEXT

$PAR
If you don't get this in $showHint tries, you can get a hint.

END_TEXT


BEGIN_HINT

$PAR
The differential equation can be rewritten as
\[ \bigl(y''\bigr)' = $Diff_Eq_r1 \]
i.e.
\[ x_3' = $Diff_Eq_r2 \]
$PAR
Combine this with the fact that
\(x_1' = y' = x_2\), etc. to get the complete system.

END_HINT

Context()->normalStrings;

$showPartialCorrectAnswer=1;

ANS(Real("0")->cmp);ANS(Real("1")->cmp);ANS(Real("0")->cmp);
ANS(Real("0")->cmp);ANS(Real("0")->cmp);ANS(Real("1")->cmp);
ANS(Formula("-$p3/$p0")->cmp);
ANS(Formula("-$p2/$p0")->cmp);
ANS(Formula("-$p1/$p0")->cmp);

ANS(Real("0")->cmp);
ANS(Real("0")->cmp);
ANS(Formula("$g/$p0")->cmp);

ANS(Real("$t0")->cmp);
ANS(Real("$y0")->cmp);
ANS(Real("$yp0")->cmp);
ANS(Real("$ypp0")->cmp);


ENDDOCUMENT();