Difference between revisions of "GeneralSolutionODE1"
Paultpearson (talk | contribs) (New custom answer checker for ODEs example that also uses PGML) |
(add historical tag and give links to newer problems.) |
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+ | {{historical}} |
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+ | |||
+ | <p style="font-size: 120%;font-weight:bold">This problem has been replaced with [https://openwebwork.github.io/pg-docs/sample-problems/DiffEq/GeneralSolutionODE.html a newer version of this problem]</p> |
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+ | |||
<h2>General Solutions to ODEs with Arbitrary Constants</h2> |
<h2>General Solutions to ODEs with Arbitrary Constants</h2> |
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<tr valign="top"> |
<tr valign="top"> |
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− | <th> PG problem file </th> |
+ | <th style="width: 50%"> PG problem file </th> |
<th> Explanation </th> |
<th> Explanation </th> |
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</tr> |
</tr> |
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DOCUMENT(); |
DOCUMENT(); |
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− | loadMacros( |
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+ | loadMacros('PGstandard.pl','MathObjects.pl','PGML.pl', |
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− | "PGstandard.pl", |
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+ | 'parserAssignment.pl','PGcourse.pl'); |
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− | "MathObjects.pl", |
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− | "PGML.pl", |
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− | "AnswerFormatHelp.pl", |
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− | "parserAssignment.pl", |
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− | "PGcourse.pl", |
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− | ); |
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TEXT(beginproblem()); |
TEXT(beginproblem()); |
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− | $showPartialCorrectAnswers = 1; |
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</pre> |
</pre> |
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</td> |
</td> |
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Line 58: | Line 55: | ||
<b>Initialization:</b> |
<b>Initialization:</b> |
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We load <code>parserAssignment.pl</code> to require student answers to be of the form <code>y = ...</code>. |
We load <code>parserAssignment.pl</code> to require student answers to be of the form <code>y = ...</code>. |
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− | Please see the POD documentation [http://webwork.maa.org/pod/ |
+ | Please see the POD documentation [http://webwork.maa.org/pod/pg/macros/parserAssignment.html parserAssignment.pl]. |
</p> |
</p> |
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</td> |
</td> |
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<td style="background-color:#ffffdd;border:black 1px dashed;"> |
<td style="background-color:#ffffdd;border:black 1px dashed;"> |
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<pre> |
<pre> |
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− | Context("Numeric"); |
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+ | 'MathObjects.pl','PGML.pl', |
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+ | 'parserAssignment.pl','PGcourse.pl'); |
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+ | |||
+ | TEXT(beginproblem()); |
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+ | $showPartialCorrectAnswers = 1; |
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+ | |||
+ | Context('Numeric'); |
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Context()->variables->add( |
Context()->variables->add( |
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− | + | c1=>'Real',c2=>'Real',c3=>'Real',y=>'Real', |
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); |
); |
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Context()->variables->set( |
Context()->variables->set( |
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− | + | c1=>{limits=>[2,4]}, |
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− | + | c2=>{limits=>[2,4]}, |
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− | + | c3=>{limits=>[2,4]} |
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); |
); |
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Context()->flags->set( |
Context()->flags->set( |
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# char poly (r-1)(r^2 + $a) |
# char poly (r-1)(r^2 + $a) |
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− | |||
− | $diffeq = "y^{\,\prime\prime\prime} - y^{\,\prime\prime} + $a y^{\,\prime} - $a y = 0"; # tex |
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$answer = Compute("y = c1 e^x + c2 cos(sqrt($a) x) + c3 sin(sqrt($a) x)"); |
$answer = Compute("y = c1 e^x + c2 cos(sqrt($a) x) + c3 sin(sqrt($a) x)"); |
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− | </pre> |
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− | </td> |
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− | <td style="background-color:#ffffcc;padding:7px;"> |
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− | <p> |
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− | <b>Setup:</b> |
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− | Add the arbitrary constants <code>c1, c2, c3</code> to the context as variables so that we can |
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− | evaluate them later. Set the domain of function evaluation on these variables to something |
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− | sensible. Use <code>parser::Assignment->Allow;</code> to allow equation answers of the form |
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− | <code>y = ...</code>. |
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− | </p> |
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− | </td> |
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− | </tr> |
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− | <!-- Main text section --> |
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+ | $cmp = $answer->cmp( checker => sub { |
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− | |||
− | <tr valign="top"> |
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− | <td style="background-color:#ffdddd;border:black 1px dashed;"> |
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− | <pre> |
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− | BEGIN_PGML |
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− | Find the general solution to [` [$diffeq] `]. |
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− | In your answer, use [` c_1, c_2 `] and [` c_3 `] to |
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− | denote arbitrary constants and [` x `] |
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− | the independent variable. Your answer should |
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− | be an equation of the form [`y = \ldots`] and |
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− | you should enter |
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− | [` c_1 `] as [| c1 |]*, |
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− | [` c_2 `] as [| c2 |]*, and |
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− | [` c_3 `] as [| c3 |]*. |
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− | |||
− | [_________________________________] |
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− | [@ AnswerFormatHelp("equations" @]* |
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− | END_PGML |
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− | </pre> |
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− | <td style="background-color:#ffcccc;padding:7px;"> |
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− | <p> |
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− | <b>Main Text:</b> |
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− | Give students detailed instructions about the format |
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− | of the answer that is expected. |
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− | </p> |
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− | </td> |
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− | </tr> |
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− | |||
− | <!-- Answer evaluation section --> |
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− | |||
− | <tr valign="top"> |
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− | <td style="background-color:#eeddff;border:black 1px dashed;"> |
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− | <pre> |
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− | ANS( $answer->cmp( checker => sub { |
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my ( $correct, $student, $answerHash ) = @_; |
my ( $correct, $student, $answerHash ) = @_; |
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# Check for arbitrary constants |
# Check for arbitrary constants |
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# |
# |
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− | Value->Error("Is your answer the most general solution?") |
+ | Value->Error("Is your answer the most general solution?") |
− | if ( |
+ | if ( |
− | Formula($stu->D('c1'))==Formula(0) || |
+ | Formula($stu->D('c1'))==Formula(0) || |
− | Formula($stu->D('c2'))==Formula(0) || |
+ | Formula($stu->D('c2'))==Formula(0) || |
− | Formula($stu->D('c3'))==Formula(0) |
+ | Formula($stu->D('c3'))==Formula(0) |
); |
); |
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− | |||
− | ################################## |
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# Linear independence (Wronskian) |
# Linear independence (Wronskian) |
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− | # |
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my $x = Real(1.43); |
my $x = Real(1.43); |
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my $a33 = $stu->D('x','x')->eval('c1'=>0,'c2'=>0,'c3'=>1,x=>$x,y=>0); |
my $a33 = $stu->D('x','x')->eval('c1'=>0,'c2'=>0,'c3'=>1,x=>$x,y=>0); |
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− | # my $wronskian = |
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+ | Value->Error("Your functions are not linearly independent or your answer is not complete") |
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− | + | if (($a11*($a22*$a33-$a32*$a23)+$a13*($a21*$a32-$a31*$a22)) == ($a12*($a21*$a33-$a31*$a23))); |
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− | # $a12*($a21*$a33-$a31*$a23) + |
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− | # $a13*($a21*$a32-$a31*$a22); |
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− | # Value->Error("Your functions are not linearly independent or your answer is not complete") |
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− | # if ($wronskian==Real(0)); |
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− | Value->Error("Your functions are not linearly independent or your answer is not complete") |
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− | if (($a11*($a22*$a33-$a32*$a23)+$a13*($a21*$a32-$a31*$a22)) == ($a12*($a21*$a33-$a31*$a23))); |
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− | |||
− | ################################# |
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# Check that the student answer is a solution to the DE |
# Check that the student answer is a solution to the DE |
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− | # |
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my $stu1 = Formula($stu->D('x')); |
my $stu1 = Formula($stu->D('x')); |
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my $stu2 = Formula($stu->D('x','x')); |
my $stu2 = Formula($stu->D('x','x')); |
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return ($stu3 + $a * $stu1) == ($stu2 + $a * $stu); |
return ($stu3 + $a * $stu1) == ($stu2 + $a * $stu); |
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− | # my $stuDE = Formula($stuxxx - $stuxx + $a*$stux - $a*$stu) |
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+ | }); |
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− | # ->with(test_points=>[[1,1,2,0.1,0],[2,1,1,0,0],[1,2,-1,-0.1,0],[-2,1,1,0,1]]); |
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− | # return ($stuDE==Formula(0)); |
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− | |||
− | |||
− | })); |
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− | |||
− | COMMENT("MathObject version. Characteristic polynomial (r-1)(r^2 + a). Uses PGML."); |
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− | |||
− | ENDDOCUMENT(); |
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</pre> |
</pre> |
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− | <td style="background-color:#eeccff;padding:7px;"> |
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+ | </td> |
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+ | <td style="background-color:#ffffcc;padding:7px;"> |
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<p> |
<p> |
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− | <b> |
+ | <b>Setup:</b> |
− | + | Add the arbitrary constants <code>c1, c2, c3</code> to the context as variables so that we can |
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+ | evaluate them later. Set the domain of function evaluation on these variables to something |
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+ | sensible. Use <code>parser::Assignment->Allow;</code> to allow equation answers of the form |
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+ | <code>y = ...</code>. |
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+ | </p> |
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+ | <p>For the <code>checker</code>, we use <code>my $stu = Formula($student->{tree}{rop});</code> to get the right side of the |
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student answer (to get the left side, we could have used <code>lop</code> for the left operand). |
student answer (to get the left side, we could have used <code>lop</code> for the left operand). |
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Use <code>Formula($stu->D('c1'))==Formula(0)</code> to check that the student actually has |
Use <code>Formula($stu->D('c1'))==Formula(0)</code> to check that the student actually has |
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in order to get a more reliable answer checker. |
in order to get a more reliable answer checker. |
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</p> |
</p> |
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+ | |||
+ | |||
</td> |
</td> |
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</tr> |
</tr> |
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+ | |||
+ | <!-- Main text section --> |
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+ | |||
+ | <tr valign="top"> |
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+ | <td style="background-color:#ffdddd;border:black 1px dashed;"> |
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+ | <pre> |
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+ | BEGIN_PGML |
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+ | Find the general solution to |
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+ | [` y^{\,\prime\prime\prime} - y^{\,\prime\prime} + [$a] y^{\,\prime} - [$a] y = 0 `]. |
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+ | |||
+ | In your answer, use [` c_1, c_2 `] and [` c_3 `] to denote arbitrary constants and [` x `] |
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+ | the independent variable. Your answer should be an equation of the form [`y = \ldots`] and |
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+ | you should enter [` c_1 `] as [| c1 |]*, |
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+ | [` c_2 `] as [| c2 |]*, and |
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+ | [` c_3 `] as [| c3 |]*. |
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+ | |||
+ | [_________________________________]{$cmp} |
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+ | |||
+ | [@ helpLink('equations') @]* |
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+ | END_PGML |
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+ | |||
+ | </pre> |
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+ | <td style="background-color:#ffcccc;padding:7px;"> |
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+ | <p> |
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+ | <b>Main Text:</b> |
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+ | Give students detailed instructions about the format |
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+ | of the answer that is expected. |
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+ | </p> |
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+ | </td> |
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+ | </tr> |
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+ | <!-- Solution section --> |
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+ | |||
+ | <tr valign="top"> |
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+ | <td style="background-color:#ddddff;border:black 1px dashed;"> |
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+ | <pre> |
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+ | BEGIN_PGML_SOLUTION |
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+ | Solution explanation goes here. |
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+ | END_PGML_SOLUTION |
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+ | |||
+ | ENDDOCUMENT(); |
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+ | </pre> |
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+ | <td style="background-color:#ddddff;padding:7px;"> |
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+ | <p> |
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+ | <b>Solution:</b> |
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+ | </p> |
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+ | </td> |
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+ | </tr> |
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+ | |||
</table> |
</table> |
Latest revision as of 06:27, 18 July 2023
This problem has been replaced with a newer version of this problem
General Solutions to ODEs with Arbitrary Constants
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
.
- PGML location in OPL: FortLewis/Authoring/Templates/DiffEq/GeneralSolutionODE1_PGML.pg
PG problem file | Explanation |
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Problem tagging: |
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DOCUMENT(); loadMacros('PGstandard.pl','MathObjects.pl','PGML.pl', 'parserAssignment.pl','PGcourse.pl'); TEXT(beginproblem()); |
Initialization:
We load |
'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 * $stu); }); |
Setup:
Add the arbitrary constants For the
We substitute numerical values that the student is unlikely to choose for Finally, we take several derivatives of the student answer and use them to check that the student answer actually satisfies the differential equation. Again, instead of checking (left side of ODE) == 0, we rearrange the terms of the differential equation to be of the form (some nonzero function) == (some other nonzero function) in order to get a more reliable answer checker.
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BEGIN_PGML Find the general solution to [` y^{\,\prime\prime\prime} - y^{\,\prime\prime} + [$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: |