impulse-response.pg

## DESCRIPTION
## Systems of ODEs: Nonhomogeneous Systems
## ENDDESCRIPTION

## DBsubject(Differential equations)
## DBchapter(Higher order differential equations)
## DBsection(Variation of parameters)
## Date(24/11/2015)
## Institution(METU-NCC)
## Author(Benjamin Walter)


##################################
#  Initialization

DOCUMENT();
loadMacros(
  "PGstandard.pl",
  "MathObjects.pl",
  "PGmatrixmacros.pl",
  "parserMultiAnswer.pl",
  "answerHints.pl",
);


#####################################
#  Setup

Context("Numeric");
Context()->variables->{namePattern} = qr/[a-z][a-z0-9_]*'*/i;
Context()->variables->are(
   t=>"Real",
   r=>"Real",
   y=>"Real", "y'"=>"Real", "y''"=>"Real",
);
Context()->flags->set(
   zeroLevel=>0.001,
   limits=>[0,2],
   zeroLevelTol=>0.001
);



############################
# setup differential equation 

my $a = random(-5,5,1);     my $b;

do { $b = random(-5,5,1); } until ($a != $b);  # roots of char. eqn.

$char_eqn = Formula("r^2 - ($a+$b) r  + ($a*$b)  ")->reduce;
       $L = Formula("y'' - ($a+$b) y' + ($a*$b) y")->reduce;  # DE

my $c = non_zero_random(-9,9,1);
   $g = Formula("$c*t^2*e^{$a*t}")->reduce;    # forcing function
            
my $c1 = non_zero_random(-3,3,1);   my $c2;
if ($b != 0) {
  $c2 = random(max(int ((-9-$a*$c1)/$b),-10), min(int ((9-$a*$c1)/$b),10), 1); 
} else {
  $c2 = random(int (-9/abs($a)), int (9/abs($a)), 1); 
}

$y0 =     $c1 +    $c2;
$dy0 = $a*$c1 + $b*$c2;

#############################
# answer evaluators

# A -- homogeneous, initial values

$IVP = Formula("$c1*e^($a*t) + $c2*e^($b*t)")->reduce; 

# B -- homogeneous, impulse response 

$IR = Formula("(e^($a*t) - e^($b*t))/($a-$b)")->reduce;

# C -- forcing function

#  Main text

Context()->texStrings;
BEGIN_TEXT
Solutions to linear differential equations can be written using convolutions 
as
\[ y \ \  = \ \ y_{\mathrm{IVP}} \ \ + \ \ \Bigl(h(t) \ {\Large \ast} \ f(t)\Bigr) \] 
\( \qquad \bullet\) 
 \( y_{\mathrm{IVP}} \) is the solution to the 
 associated homogeneous differential equation with the given initial values 
$BR \(\qquad\qquad\) (ignore the forcing function, keep initial values).
$BR
\( \qquad \bullet\) 
 \( h(t) \) is the ${BTT}impulse response${ETT}
$BR \(\qquad\qquad\) (ignore the initial values and forcing function).
$BR
\( \qquad \bullet\)
 \( f(t) \) is the forcing function.
$BR \(\qquad\qquad\) (ignore the initial values and differential equation).

$PAR
$HR
$PAR

Use the form above to write the solution to the differential equation
\[ $L = $g \qquad \mathrm{with} \quad y(0) = $y0, \quad y'(0) = $dy0 \]

$PAR

\( y \ \ = \ \ \)\{ans_rule(10)\}\( \quad + \quad 
  \Bigl(\)\{ans_rule(10)\}\(\!\Large\ast\)\{ans_rule(10)\}\(\Bigr)\)

END_TEXT

$showHint = 1;

BEGIN_TEXT
$BR
$HR
$PAR
If you don't get this in $showHint tries, you can get a hint.
END_TEXT
BEGIN_HINT
You can quickly compute the impulse response by converting the impulse at \(t=0\)
to an initial velocity.  Solve
\[ $L = 0,\qquad \mathrm{with}\quad y(0) = 0, \quad y'(0) = 1\]
using the characteristic equation 
\[ $char_eqn = 0 \]
and plugging in to compute \(c_1\) and \(c_2\). 
(You could also compute using Laplace transforms, but that is more slow.)
$PAR
$HR
$PAR
To be accepted, your answer must be entered into webwork as $BR
$SPACE $SPACE ${BTT}(Impulse resp.)${ETT}\(\Large\ast\)${BTT}(Forcing fun.)${ETT} $BR
${BITALIC}Hint:${EITALIC} The forcing function is \(f(t) = $g\).
END_HINT

Context()->normalStrings;

#####################################
#  Answer evaluation

$showPartialCorrectAnswers = 2;

ANS( $IVP->cmp() );
ANS( $IR->cmp() );
ANS( $g->cmp() );

COMMENT('MathObject version. Randomized -- DISTINCT roots.');

ENDDOCUMENT();