linear-nonhomog_euler.pg

## DESCRIPTION
## Systems of ODEs: Nonhomogeneous Systems
## ENDDESCRIPTION

## DBsubject(Differential equations)
## DBchapter(Higher order differential equations)
## DBsection(Variation of parameters)
## Date(24/11/2015)
## Institution(METU-NCC)
## Author(Benjamin Walter)


##################################
#  Initialization

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


#####################################
#  Setup

Context("Numeric");
Context()->variables->{namePattern} = qr/[a-z][a-z0-9_]*'*/i;
Context()->variables->are(
   t=>["Real", limits=>[1,3]], 
   n=>"Real",
   y=>"Real", "y'"=>"Real", "y''"=>"Real",
   "c1"=>"Real", "c2"=>"Real"
);
Context()->flags->set(
   zeroLevel=>0.001,
   limits=>[0,2],
   zeroLevelTol=>0.001
);



############################
# setup differential equation 

my $a1, $a2, $b1, $b2, $c, $d;  #  roots are a1/b1 and a2/b2
my @coeffs;

$b1 = 1; # random(1,3,1);  
$b2 = 1;
$a1 = non_zero_random($b1-4, 4-$b1, 1)*$b1 
     + (($b1 == 1) ? 0 : non_zero_random(1-$b1,$b1-1,1));
do { $a2 = non_zero_random(-3,3,1); }
until ( $a2/$b2 != $a1/$b1 );

$c = non_zero_random(-3,3,1);  $d = random(-3,3,1);

@coef = (Formula("$b1*$b2"), 
         Formula("$b1*$b2-$a1*$b2-$a2*$b1"),
         Formula("$a1*$a2"));

$L = Formula("$coef[0]*t^2*y'' + $coef[1]*t*y' + $coef[2]*y")->reduce;  # DE

$g = Formula("$coef[0]*$c*t^3 + $coef[0]*$d*t^2")->reduce;   # forcing function
$real_g = Formula("$c*t+$d")->reduce;

#############################
# answer evaluators

# A -- accept any multiple of eqn (rounding error => only small multiples)

$eqn = Formula("(t^n)*($b1*n-$a1)*($b2*n-$a2)")->reduce;
$eqn_hint1 = Formula("$coef[0]*t^2*(n*(n-1)*t^(n-2)) + $coef[1]*t*(n*t^(n-1)) + $coef[2]*(t^n)")->reduce;
$eqn_hint2 = Formula("t^n($coef[0]*n*(n-1) + $coef[1]*n + $coef[2])")->reduce;
$eqn_hint25 = Formula("($coef[0]*n*(n-1) + $coef[1]*n + $coef[2])")->reduce;
$eqn_hint3 = Formula("$coef[0]*n^2 + ($coef[1]-$coef[0])*n + $coef[2]")->reduce;
$eqn_hint4 = Formula("($b1*n-$a1)*($b2*n-$a2)")->reduce;

# B -- fundamental solutions and wronskian

$y1 = Formula("t^($a1/$b1)")->reduce;   
$y2 = Formula("t^($a2/$b2)")->reduce;
my $Wcoef = Formula("(($a2*$b1-$a1*$b2)/($b1*$b2))");
my $Wronskian = Formula("$Wcoef*t^(($a2*$b1+$a1*$b2-$b1*$b2)/($b1*$b2))")->reduce;

my @S_fund = (Formula("$y1"), 
              Formula("$y2"));            # record student solutions
my $S_wron = Formula("$Wronskian");       # record student wronskian 

$multiansB = MultiAnswer($y1, $y2, $Wronskian)->with(
    singleResult => 1,
    checkTypes => 0,
    allowBlankAnswers => 1,
    format =>
    "<table border='0' cellspacing='0'>
     <tr><td>\(y_1 = \) %s </td></tr>
     <tr><td>\(y_2 = \) %s </td></tr>
     <tr><td>\(W = \) %s </td></tr>
    </table>",
    tex_format =>
    "\begin{array}{l} y_1 = %s \\ y_2 = %s \\ W = %s
     \end{array}",

    checker => sub {
        my ($correct, $student, $answerHash ) = @_;
        my @c = @{$correct};
        my @s = @{$student};
        my $msg = "";   my @flag = (0,0,0);   my @S=(0,0);

        if ($c[0]->typeMatch($s[0]) &&
            $c[1]->typeMatch($s[1])) {  
          if ((Formula("$s[0]")->usesOneOf("c1","c2") ||
               Formula("$s[1]")->usesOneOf("c1","c2")) &&
             !(Formula("$s[2]")->usesOneOf("c1","c2"))) {
            Value::Error("If \(y_1\) and \(y_2\) use \(c_1\) and \(c_2\), then so does \(W(y_1,y_2)\)!${BR}(Easier if \(y_1\) and \(y_2\) don't have \(c_1\) and \(c_2\)...)");
          }
          @S = ( Formula("$s[0]"),
                 Formula("$s[1]") );
          my @dS = ( Formula("$s[0]")->D('t'),
                     Formula("$s[1]")->D('t') );
          my @ddS = ( Formula("$s[0]")->D('t','t'),
                      Formula("$s[1]")->D('t','t') );
          $S_wron = Formula("$S[0]*$dS[1] - $S[1]*$dS[0]")->reduce;
          if ($S_wron == Formula("0")) {
            Value::Error("Fundmental solutions not independent!");
          }
          my @L = ( Formula("$coef[0]*t^2*$ddS[0] + $coef[1]*t*$dS[0] + $coef[2]*$S[0]")->reduce, 
                    Formula("$coef[0]*t^2*$ddS[1] + $coef[1]*t*$dS[1] + $coef[2]*$S[1]")->reduce );
          if ($L[0] == Formula("0")) { $flag[0] = 1; $msg = "\(y_1\) is fine$BR"; }
          else  { $msg = "\(y_1\) is not a fundamental solution$BR"; }

          if ($L[1] == Formula("0")) { $flag[1] = 1; $msg = $msg . "\(y_2\) is fine$BR"; }
          else  { $msg = $msg . "\(y_2\) is not a fundamental solution$BR"; }
        }
  
        if (($flag[0]*$flag[1]) == 1) {
          $S_fund[0] = Formula("$s[0]");
          $S_fund[1] = Formula("$s[1]");
        }

        if ($c[2]->typeMatch($s[2]) && 
           ($S_wron == Formula("$s[2]"))) { $flag[2]=1; $S_wron = Formula("$s[2]")->reduce; }
        else { Value::Error($msg . " Wronskian is incorrect!"); }

        if ($flag[0]*$flag[1]*$flag[2] == 1) { return 1; }
        else { if (length($msg) > 1) { Value::Error($msg); } }
        
        return 0;
    }
);

# C - int fund * g / wronskian dt

my $iFgt1, $iFgt2, $iFgt3, $iFgt4;

$iFgt1=(($a2/$b2 == 3) ? Formula("ln|t|") : Formula("($b2/(3*$b2-$a2))*t^((3*$b2-$a2)/$b2)"));
$iFgt2=(($a2/$b2 == 2) ? Formula("ln|t|") : Formula("($b2/(2*$b2-$a2))*t^((2*$b2-$a2)/$b2)"));
$iFgt3=(($a1/$b1 == 3) ? Formula("ln|t|") : Formula("($b1/(3*$b1-$a1))*t^((3*$b1-$a1)/$b1)"));
$iFgt4=(($a1/$b1 == 2) ? Formula("ln|t|") : Formula("($b1/(2*$b1-$a1))*t^((2*$b1-$a1)/$b1)"));

my @iFg = ( Formula("($c*$iFgt1+$d*$iFgt2)/($Wcoef)")->reduce,
            Formula("($c*$iFgt3+$d*$iFgt4)/($Wcoef)")->reduce );

$multiansD = MultiAnswer($iFg[0], $iFg[1])->with(
    singleResult => 1,
    checkTypes => 0,
    allowBlankAnswers => 1,
    format =>
    "<table border='0' cellspacing='0'>
     <tr><td>\(\int \frac{y_1 g}{W} dt = \) %s </td></tr>
     <tr><td>\(\int \frac{y_2 g}{W} dt = \) %s </td></tr>
    </table>",
    tex_format =>
    "\begin{array}{l} 
        \displaystyle \int \frac{y_1 g}{W} dt  = %s \\ 
        \displaystyle \int \frac{y_2 g}{W} dt  = %s 
     \end{array}",

    checker => sub {
        my ($correct, $student, $answerHash ) = @_;
        my @c = @{$correct};
        my @s = @{$student};
        my $msg = "";    my @flag = (0,0); 
        if ($c[0]->typeMatch($s[0]) &&
            $c[1]->typeMatch($s[1])) {  
          my @dS = (Formula("$s[0]")->D('t'),
                    Formula("$s[1]")->D('t'));
          if (Formula("$dS[0]*$S_wron") == Formula("$S_fund[0]*$real_g")) {
            $flag[0] = 1;
            $msg = "First integral is correct $BR";
          } else {
            $msg = "First integral is incorrect! $BR";
          } if (Formula("$dS[1]*$S_wron") == Formula("$S_fund[1]*$real_g")) {
            $flag[1] = 1;
            $msg = $msg . "Second integral is correct";
          } else {
            my $tmp1 = Formula("$s[1]")->D('t')->reduce; 
            my $tmp2 = Formula("$sFw[1]*$g")->reduce;
            $msg = $msg . "Second integral is incorrect!";
          }
        }
        if ($flag[0]*$flag[1] == 1) { return 1; }
        else { if (length($msg)>1) { Value::Error($msg); } }
        
        return 0;
    }
);

# D - general solution

$gen_soln = Formula("($iFg[0]+c2)*$y2 - ($iFg[1]+c1)*$y1")->reduce;

$multiansE = MultiAnswer($gen_soln)->with(
    singleResult => 1,
    checkTypes => 1,
    allowBlankAnswers => 0,
 
    checker => sub{
       my ($correct, $student, $answerHash ) = @_;
       my @s = @{$student};
       my $msg = "";    my @flag = (0,0,0);  

       my @S_h = (Formula("$s[0]")->D("c1"),
                  Formula("$s[0]")->D("c2"));
       my @dS_h = ($S_h[0]->D('t'),
                   $S_h[1]->D('t'));
       my @ddS_h = ($S_h[0]->D('t','t'),
                    $S_h[1]->D('t','t'));
       if (Formula("$S_h[0]*$dS_h[1]") == Formula("$S_h[1]*$dS_h[0]")) {
          Value::Error("Not the most general answer...");
       } 
       if (Formula("$coef[0]*t^2*$ddS_h[0] + $coef[1]*t*$dS_h[0] + $coef[2]*$S_h[0]") == Formula("0")) {
          $msg = "\(y_1\) is okay$BR";
          $flag[0] = 1;
       } else {
          $msg = "\(y_1\) is wrong$BR";
       }
       if (Formula("$coef[0]*t^2*$ddS_h[1] + $coef[1]*t*$dS_h[1] + $coef[2]*$S_h[1]") == Formula("0")) {
          $msg = $msg . "\(y_2\) is okay$BR";
          $flag[1] = 1;
       } else {
          $msg = $msg . "\(y_2\) is wrong$BR";
       }

       my $S_p = Formula("$s[0]")->substitute("c1"=>"0","c2"=>"0");
       my $dS_p = $S_p->D('t');
       my $ddS_p = $S_p->D('t','t');
       my $P = Formula("$coef[0]*t^2*$ddS_p + $coef[1]*t*$dS_p + $coef[2]*$S_p");
       if ($P == Formula("$g")) {
          $msg = $msg . "Particular solution is okay";
          $flag[2] = 1;
       } else {
          $msg = $msg . "Not a solution to the DE!";
       }
      
       if ($flag[0]*$flag[1]*$flag[2] != 1) {
         if (length($msg) > 1) { Value::Error($msg); }
       } else {
         return 1;
       }
       return 0;
    }
);


#####################################
#  Main text

Context()->texStrings;
BEGIN_TEXT

$Message
$PAR

In this problem you will use variation of parameters to solve the nonhomogeneous equation 
\[ $L = $g \]

$PAR
$HR
$PAR

$BBOLD A. $EBOLD
Plug \(y=t^n\) into the associated ${BBOLD}homogeneous${EBOLD} equation (with
"\(0\)" instead of "\($g\)") to get an equation with only \(t\) and \(n\).
\{ mbox([
 "\(\quad\)",
 ans_rule(40),
]) \}
(Note: Do not cancel out the \(t\), or webwork won't accept your answer!)

$PAR

$BBOLD B. $EBOLD
Solve the equation above for \(n\) (use \(t\neq 0\) to cancel out the \(t\)).
$BR $SPACE $SPACE 
You should get two values for \(n\), which give two fundamental solutions 
of the form \(y = t^n\).
\{ mbox([
  "\( \quad \phantom{\mathrm{W}(y_1,\,)} y_1 = \)",
  $multiansB->ans_rule(10),
  "\(\qquad y_2 = \)",
  $multiansB->ans_rule(10)
]) \}
\{ mbox([
  "\(\quad \mathrm{W}\bigl(y_1, y_2\bigr) = \)",
  $multiansB->ans_rule(20)
]) \}

$PAR

$BBOLD C. $EBOLD
To use variation of parameters, the linear differential equation must be written in 
standard form \(y'' + py' + qy = g\).  
$BR $SPACE $SPACE 
What is the function \(g\)?
\{ mbox ([
  "\( \qquad g(t) = \)",
  ans_rule(10)
]) \}

$PAR

$BBOLD D. $EBOLD
Compute the following integrals.
\{ mbox ([
  "\( \quad \displaystyle \int \frac{y_1 g}{W} dt = \)",
  $multiansD->ans_rule(35)
]) \}
\{ mbox ([
  "\( \quad \displaystyle \int \frac{y_2 g}{W} dt = \)",
  $multiansD->ans_rule(35)
]) \}

$PAR

$BBOLD E. $EBOLD
Write the general solution.
(Use ${BTT}c1${ETT} and ${BTT}c2${ETT} for \(c_1\) and \(c_2\)).
\{ mbox ([
  "\( \quad y = \)",
  $multiansE->ans_rule(70)
]) \}


END_TEXT

$showHint = 3;

BEGIN_TEXT
$HR
$PAR
If you don't get this in $showHint tries, you can get a hint to help you 
find the fundamental solutions.
END_TEXT

BEGIN_HINT
$PAR
$SPACE $SPACE \(\bullet\) The first two parts go as follows:
\[ \begin{array}{lrl}
  \text{homogeneous } \rlap{\text{equation:}} 
         & $L  &= \ \ 0  \\
  \text{plug in }y=t^n: \quad 
         & $eqn_hint1 &= \ \ 0 \\
  \text{factor out }t^n: 
         & $eqn_hint2 &= \ \ 0 \\
  \text{cancel }t^n: 
         & $eqn_hint25 &= \ \ 0 \\
  \text{simplify}: 
         & $eqn_hint3 &= \ \ 0 \\
  \text{factor}: 
         & $eqn_hint4 &= \ \ 0 \\
  \end{array} \]
\[ \text{fundamental solutions:}
   \quad y_1 = $y1, \quad y_2 = $y2  \]
$PAR
$SPACE $SPACE \(\bullet\) You are on your own for the rest. ;)

END_HINT


Context()->normalStrings;

#####################################
#  Answer evaluation

$showPartialCorrectAnswers = 1;

ANS( equation_cmp("$eqn = 0", vars=>['t','n']) );
ANS( $multiansB->cmp() );
ANS( $real_g->cmp() );
ANS( $multiansD->cmp() );
ANS( $multiansE->cmp() );

COMMENT("MathObject version. Randomized -- currently integer powers (rational powers don't check right).");

ENDDOCUMENT();