bvp.pg

## DBsubject(Differential equations)
## Date(20/12/2015)
## Institution(METU-NCC)
## Author(Benjamin Walter)
## Level(5)
## TitleText1('Differential Equations')
## KEYWORDS('ODE', 'boundary value')
########################################################################

DOCUMENT();      

loadMacros(
   "PGstandard.pl",   
   "MathObjects.pl",
   "answerHints.pl",
   "parserMultiAnswer.pl",
   "unionLists.pl",
   "PGchoicemacros.pl",
   "contextFraction.pl"
);

TEXT(beginproblem());

$showPartialCorrectAnswers = 1;
##############################################################
#
#  Setup
#
#
Context("Numeric");
Context()->variables->{namePattern} = qr/[a-z][a-z0-9_]*'*/i;
Context()->variables->are(
  "y"=>"Real","y'"=>"Real","y''"=>"Real");
$DEcontext = Context()->copy;

Context("Numeric");
Context()->variables->are(
  "x"=>"Real",
  "n"=>["Real",limits=>[1,6],resolution=>1],
  "a"=>"Real","b"=>"Real","c"=>"Real","d"=>"Real");
Context()->strings->add(
  "No Solution"=>{alias=>"DNE"},
  "No Solutions"=>{alias=>"DNE"});
$Default = Context()->copy;

###########
##
## Differential equation

my $beta = random(1,5,1);           # frequency for solutions

my @n = (random(1,min($beta*3,10),1),  # right boundary time  
         random(1,min($beta*3,10),1),  #  $t = $n (pi / $beta) 
         random(0,min($beta*3,10),1)); # (2$n+1)/2 (pi / $beta)

Context("Fraction");  # right boundary times $t = $a/$b
my @a,@b;             #  -- try to convert to pretty fractions
@t = (Fraction("($n[0])/$beta"),
      Fraction("($n[1])/$beta"),
      Fraction("((2*$n[2]+1))/(2*$beta)")); 
for(my $i=0;$i<3;$i++) {($a[$i],$b[$i]) = $t[$i]->value;}
@t = (Formula("$a[0]*pi/$b[0]")->reduce,
      Formula("$a[1]*pi/$b[1]")->reduce,
      Formula("$a[2]*pi/$b[2]")->reduce);
Context("Numeric");

$y0 = random(-3,3,1);               # left boundary value y(0)
if ($y0 == 0) {
  @BV = (0,
         non_zero_random(-3,3,1),
         random(-3,3,1));
} else {
  @BV = (($n[0]%2==0) ?  $y0 : -$y0,
         ($n[1]%2==0) ? -$y0 :  $y0,
         non_zero_random(-1,1,1)*$y0);
}

my $s = ($n[2]%2==0) ? 1 : -1;       # is sin($t[2]*$beta) = 1 or -1 ?

$DE = Formula($DEcontext, "y'' + $beta^2 y")->reduce;

@soln = (Formula($Default, "$y0*cos($beta x) + c sin($beta x)"),
         Formula($Default, "DNE"),
         Formula($Default, "$y0*cos($beta x) + $s*$BV[2]*sin($beta x)"));


@mix = NchooseK(3,3);

###########
##
##  Text

Context()->texStrings;
BEGIN_TEXT

Solve the following differential equation with the given boundary 
conditions.  
$BR $SPACE $SPACE
- If there are infinitely many solutions, use ${BBOLD}c${EBOLD} for any 
undetermined constants. 
$BR $SPACE $SPACE
- If there are no solutions, write ${BBOLD}No Solution${EBOLD}.  
$BR $SPACE $SPACE
- Write answers as functions of \(x\) (i.e. \(y=y(x)\)).

$PAR

\[\displaystyle $DE = 0 \]

$PAR

\{ BeginList('UL') \}

$ITEM
$BBOLD A) $EBOLD $SPACE $SPACE  Boundary conditions:
  \(\displaystyle \quad \begin{aligned}[t] 
                           y(0) &= $y0 \\
      y\left($t[$mix[0]]\right) &= $BV[$mix[0]]
                         \end{aligned} \)
$PAR $SPACE $SPACE
\(y = \) \{ ans_rule(40) \}

$ITEMSEP

$ITEM 
$BBOLD B) $EBOLD $SPACE $SPACE  Boundary conditions:  
  \(\displaystyle \quad \begin{aligned}[t] 
                           y(0) &= $y0 \\
      y\left($t[$mix[1]]\right) &= $BV[$mix[1]]
                         \end{aligned} \)
$PAR $SPACE $SPACE
\(y = \) \{ ans_rule(40) \}
                      
$ITEMSEP

$ITEM
$BBOLD C) $EBOLD $SPACE $SPACE   Boundary conditions:
  \(\displaystyle \quad \begin{aligned}[t] 
                           y(0) &= $y0 \\
      y\left($t[$mix[2]]\right) &= $BV[$mix[2]]
                         \end{aligned} \)
$PAR $SPACE $SPACE
\(y = \) \{ ans_rule(40) \}

\{ EndList('UL') \}


END_TEXT
Context()->normalStrings;

##############
##
## Answers

ANS($soln[$mix[0]]->cmp);
ANS($soln[$mix[1]]->cmp);
ANS($soln[$mix[2]]->cmp);



ENDDOCUMENT();