** Using WeBWorK **

DOCUMENT();      
loadMacros("PGstandard.pl", "PGML.pl", "PGcourse.pl");
Context("Numeric");
Context()->variables->are(k1=>"Real", k2=>"Real", t=>"Real");
$y1 = Compute("e^(-4t)");
$y2 = Compute("e^(-2t)");
$yp = Compute("7/85 cos(t) + 6/85 sin(t)");
$y_gen = Compute("k1 $y1 + k2 $y2 + $yp");
$const_1 = Compute(2/17); # k1
$const_2 = Compute(-1/5); # k2
$y_spec = $y_gen->substitute(k1=>$const_1, k2=>$const_2); #decimal
#$y_spec = Compute("2/17 $y1 + -1/5 $y2 + $yp"); # decimal
#$y_spec = Compute("2/17 $y1 + -1/5 $y2 + 7/85 cos(t) + 6/85 sin(t)"); # fractions 
$y_spec->{test_points} = [ [1,1,0.005] ]; # ok at 0.00920 or greater
#$y_spec->{limits} = [0.5,5]; 
BEGIN_TEXT
(a) Find the general solution of the differential equation
\[ \frac{d^2y}{dt^2} + 6 \frac{dy}{dt} + 8 y = \cos{t}. \]
\(\quad\) \(y(t) = \) \{ans_rule(50)\} 
$PAR \(\quad\) Use "k1" and "k2" for the constants in your solution.
$PAR (b) Find the solution of the initial-value problem 
\[ \frac{d^2y}{dt^2} + 6 \frac{dy}{dt} + 8 y = \cos{t}, \quad y(0)=y'(0)=0. \]
\(\quad\) \(y(t) = \) \{ans_rule(50)\} 
END_TEXT
sub mycheck {
  my ($correct, $student, $ansHash) = @_;
  if ( $student->usesOneOf("k1") && # uses constant k1 -- not necessary?
       $student->usesOneOf("k2") && # uses constant k2 -- not necessary?
       ( ($student->substitute(k1=>1, k2=>0) == $y1 + $yp) || # one of terms is y1
         ($student->substitute(k1=>0, k2=>1) == $y1 + $yp) ) && 
       ( ($student->substitute(k1=>1, k2=>0) == $y2 + $yp) || # one of terms is y2
         ($student->substitute(k1=>0, k2=>1) == $y2 + $yp) ) && 
       $student->substitute(k1=>0, k2=>0) == $yp # probably redundant w/above...
     ) 
  { return 1; } else { return 0; } 
}
ANS($y_gen->cmp(checker=>~~&mycheck));
#ANS($y_spec->cmp);
ANS($y_spec->cmp(diagnostics=>1));
ENDDOCUMENT();        

I have three versions of $y_spec. The first is what I'd like because it minimizes copy/paste errors on my part -- I need only get $yp correct in one place. If one enters the correct function

2/17 e^(-4t) - 1/5 e^(-2t) + 7/85 cos(t) + 6/85 sin(t)

and an evaluation point is unluckily chosen in the interval [-0.0092,0.0092] then the answer is marked incorrect b/c the relative error is greater than 0.001 (I checked the interval). The answer display in this first version uses six-digit decimals. That got me thinking maybe the decimal approximation is the issue. So I put the decimal approximation version and the fraction version into Matlab and plotted the relative error -- it's larger than 0.001  on the interval [-0.0141,0.0141]. That's not identical to webwork/perl but not outlandishly different either. 

The second version of $y_spec exhibits the same behavior -- decimal display and marked incorrect. 

The third version of $y_spec displays fractions and marks the answer correct -- the relative error from ANS($y_spec->cmp(diagnostics=>1)) is on the order of 10^-12. 

So: is there something going on with keeping fewer digits than double-precision floating-point? Is this at all connected to the answer display?

Thanks in advance for your thoughts on this - Thomas