I was trying to use the "parserFormulaUpToConstant.pl" when evaluating the solution to a separable differential equation.
$fp = FormulaUpToConstant("-1/( ($coef/($exp+1))x^($exp+1) + C)")->reduce();
...
ANS($fp->cmp);
This generated the Error message
Your formula isn't linear in the arbitrary constant 'C'
Is there another Macro I should be loading and using in these cases? Thank you!
Thanks Davide, But what I'd really like is the ability to use C in an arbitrary position. Currently, if the correct answer is:
-1/(4/5 x^5 +C)
Webwork will indicate the following correct answer as incorrect:
-5/(4x^5 +C)
Of course, Webwork will accept
-5/(4x^5 +5C) as correct, but this could be frustrating for students just learning about solving differential equations (especially when taught 5C is "just another" C) . I could throw in an initial condition, but wondered if a macro had been written to evaluate functions with "non-linear" arbitrary constants.
-1/(4/5x^5 + aC)
with a custom checker that verifies that the "a" is non-zero? (Or does it really need to be -1/(4/5x^5+aC + b) for adaptive parameters a and b? I.e., should -1/(4/5x^5 + 1 + C) be considered correct?)
Something like this (untested) code:
Context()->variables->add(C=>'Real');
Context()->variables->add(a=>'Parameter');
$f = Formula("-1/((4/5)x^5 + aC")
$f->{correct_ans} = $f->substitute(a=>1)->reduce;
ANS($f->cmp(checker => sub {
my ($correct,$student,$ans) = @_;
return 0 unless $correct == $student;
return Real(Context()->variables->get("a")->{value}) != 0;
}));
A more sophisticated version of this could be packaged up as a new MathObject (like FormulaUpToConstant). I would normally write that for you, but I'm a little pressed for time at the moment, so it will have to wait. I'll add it to my list of things to do.
Davide
Thanks again Davide. I added the above code. This didn't work, as the correct answer yielded "incorrect" with the message:
Can't solve for adaptive parameters
I will understand if this is too time-consuming to consider right now. Lynda
Next suppose "the" answer is:
$answer = Formula("1/(x+C)");
which is the solution to
y' = -y^2
Make a custom answer checker that checks for two things:
1. the student's answer solves the differential equation
2. whatever the student has entered in place of C is a linear (and nonconstant) function of C
I probably do not have the most efficient way to do this, but my custom checker works like this:
ANS( $answer->cmp( checker=>sub {
my ( $correct, $student, $ansHash ) = @_;
my $dstudent = $student->D('x');
my $lhs = Formula("$dstudent");
my $rhs = Formula("-($student)^2");
my $studentC = Formula("1/$student - x");
my $dstudentC = $studentC->D('C');
my $ddstudentC = $dstudentC->D('C');
if (($lhs == $rhs) && ($ddstudentC == Formula("0")))
{ if ($dstudentC != Formula("0"))
{return 1}
else {Value->Error("Your answer is not the most general solution.")}
}
else {return 0};
} ) );
This lets students enter things like 1/(x+2C-5), if that happens to be where they ended up after solving process. There is a check in there that prevents things like 1/(x+5) from being counted as correct too. Ideally, the checker would allow 1/(x+f(C)) for any one-to-one function f that covers R, but I don't know how to check that. Looking at first and second derivatives lets us check linearity though.
Take care if you use this in problems where the answer has domain issues.