GeneralSolutionODE1

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General Solutions to ODEs with Arbitrary Constants

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This PG code shows how to write a custom answer checker for ODEs questions where the answer is an equation of the form y = c1 f1(x) + c2 f2(x) + c3 f3(x) for some arbitrary constants c1, c2, c3.


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PG problem file Explanation

Problem tagging data

Problem tagging: 

DOCUMENT();

loadMacros('PGstandard.pl','MathObjects.pl','PGML.pl',
  'parserAssignment.pl','PGcourse.pl');

TEXT(beginproblem());

Initialization: We load parserAssignment.pl to require student answers to be of the form y = .... Please see the POD documentation parserAssignment.pl. 

'MathObjects.pl','PGML.pl',
  'parserAssignment.pl','PGcourse.pl');

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

Context('Numeric');
Context()->variables->add(
  c1=>'Real',c2=>'Real',c3=>'Real',y=>'Real',
);
Context()->variables->set(
  c1=>{limits=>[2,4]},
  c2=>{limits=>[2,4]},
  c3=>{limits=>[2,4]}
);
Context()->flags->set(
    formatStudentAnswer=>'parsed',
    reduceConstants=>0,
    reduceConstantFunctions=>0,
);
parser::Assignment->Allow;

$a  = list_random(2,3,5,6,7,8);

# char poly (r-1)(r^2 + $a)

$answer = Compute("y = c1 e^x + c2 cos(sqrt($a) x) + c3 sin(sqrt($a) x)");

$cmp = $answer->cmp( checker => sub {

    my ( $correct, $student, $answerHash ) = @_;
    my $stu   = Formula($student->{tree}{rop});

    #################################
    #  Check for arbitrary constants
    #
    Value->Error("Is your answer the most general solution?")
    if (
      Formula($stu->D('c1'))==Formula(0) ||
      Formula($stu->D('c2'))==Formula(0) ||
      Formula($stu->D('c3'))==Formula(0)
    );

    #  Linear independence (Wronskian)
    my $x = Real(1.43);

    my $a11 = $stu->eval('c1'=>1,'c2'=>0,'c3'=>0,x=>$x,y=>0);
    my $a12 = $stu->eval('c1'=>0,'c2'=>1,'c3'=>0,x=>$x,y=>0);
    my $a13 = $stu->eval('c1'=>0,'c2'=>0,'c3'=>1,x=>$x,y=>0);

    my $a21 = $stu->D('x')->eval('c1'=>1,'c2'=>0,'c3'=>0,x=>$x,y=>0);
    my $a22 = $stu->D('x')->eval('c1'=>0,'c2'=>1,'c3'=>0,x=>$x,y=>0);
    my $a23 = $stu->D('x')->eval('c1'=>0,'c2'=>0,'c3'=>1,x=>$x,y=>0);

    my $a31 = $stu->D('x','x')->eval('c1'=>1,'c2'=>0,'c3'=>0,x=>$x,y=>0);
    my $a32 = $stu->D('x','x')->eval('c1'=>0,'c2'=>1,'c3'=>0,x=>$x,y=>0);
    my $a33 = $stu->D('x','x')->eval('c1'=>0,'c2'=>0,'c3'=>1,x=>$x,y=>0);

    Value->Error("Your functions are not linearly independent or your answer is not complete")
      if (($a11*($a22*$a33-$a32*$a23)+$a13*($a21*$a32-$a31*$a22)) == ($a12*($a21*$a33-$a31*$a23)));


    #  Check that the student answer is a solution to the DE
    my $stu1 = Formula($stu->D('x'));
    my $stu2 = Formula($stu->D('x','x'));
    my $stu3 = Formula($stu->D('x','x','x'));
    return ($stu3 + $a * $stu1) == ($stu2 + $a * $stu);

});

Setup: Add the arbitrary constants c1, c2, c3 to the context as variables so that we can evaluate them later. Set the domain of function evaluation on these variables to something sensible. Use parser::Assignment->Allow; to allow equation answers of the form y = ....

For the checker, we use my $stu = Formula($student->{tree}{rop}); to get the right side of the student answer (to get the left side, we could have used lop for the left operand). Use Formula($stu->D('c1'))==Formula(0) to check that the student actually has c1 in their answer.

We substitute numerical values that the student is unlikely to choose for c1, c2, c3 and then apply the Wronskian test for independence. Normally, we would check to see if the Wronskian was zero, but zero level tolerance in WeBWorK is much more stringent than non-zero level tolerance. So, we rearrange the terms of the Wronskian == 0 equation so that there are nonzero terms on both sides of the equation, and as a result we get a more reliable answer checker.

Finally, we take several derivatives of the student answer and use them to check that the student answer actually satisfies the differential equation. Again, instead of checking (left side of ODE) == 0, we rearrange the terms of the differential equation to be of the form (some nonzero function) == (some other nonzero function) in order to get a more reliable answer checker. 


BEGIN_PGML
Find the general solution to
[` y^{\,\prime\prime\prime} - y^{\,\prime\prime} + [$a] y^{\,\prime} - [$a] y = 0 `].

In your answer, use [` c_1, c_2 `] and [` c_3 `] to denote arbitrary constants and [` x `]
the independent variable.  Your answer should be an equation of the form [`y = \ldots`] and
you should enter [` c_1 `] as [| c1 |]*,
[` c_2 `] as [| c2 |]*, and
[` c_3 `] as [| c3 |]*.

[_________________________________]{$cmp}

[@ helpLink('equations') @]*
END_PGML

Main Text: Give students detailed instructions about the format of the answer that is expected. 

BEGIN_PGML_SOLUTION
Solution explanation goes here.
END_PGML_SOLUTION

ENDDOCUMENT();

Solution:

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