prob45.pg

## DESCRIPTION
## Differential Equations: Laplace Transforms
## ENDDESCRIPTION

## Tagged by cmd6a 3/14/06

## DBsubject(Differential equations)
## DBchapter(Laplace transforms)
## DBsection(Basic transformations)
## Institution(ASU)
## MO(1)
## MLT(forward_01)
## MLTleader(1)
## Level(5)

DOCUMENT() ;        # This should be the first executable line in the problem.

loadMacros(
  "PGstandard.pl",
  "PGchoicemacros.pl",
  "PGgraphmacros.pl",
  "PGasu.pl",
  "MathObjects.pl",
  "PGcourse.pl"
);

TEXT(beginproblem()) ;
$showPartialCorrectAnswers = 1;

$r = random(2,6,1);
$r2 = 2*$r;
$b = random(14,19,1);

$ans5= FEQ("tanh(s/4)/(s^2*(s^2+$r2*s+$b))");
$graph3 = init_graph(-1,-2,3,2, 'axes'=>[0,0],'grid'=>[4,4]);
$f1 = "x for x in <0,0.5> using color:red and weight:2";
$f2 = "1-x for x in <0.5,1> using color:red and weight:2";
$f3 = "x-1 for x in <1,1.5> using color:red and weight:2";
$f4 = "2-x for x in <1.5,2> using color:red and weight:2";
$f5 = "x-2 for x in <2,2.5> using color:red and weight:2";
$f6 = "3-x for x in <2.5,3> using color:red and weight:2";
plot_functions($graph3, $f1);
plot_functions($graph3, $f2);
plot_functions($graph3, $f3);
plot_functions($graph3, $f4);
plot_functions($graph3, $f5);
plot_functions($graph3, $f6);
$g = image(insertGraph($graph3),
    alt=>"Triangular wave function with period 1, showing repeated peaks at height 0.5 occurring at t = 0.5, 1.5, and 2.5, with vertices touching the x-axis at integer values."
);

$func = "\begin{cases}
            t, & 0 \leq t < 1/2 \cr
            1-t, & 1/2 \leq t < 1
             \end{cases}";

BEGIN_TEXT
Take the Laplace transform of the following initial value problem and 
solve for \(Y(s) = \mathcal{L}$LBRACE y(t) $RBRACE \):

\[ y'' + {$r2} y' + {$b} y= T(t) \hspace{0.5in} y(0)=0, \; y'(0)=0\]
$PAR  
Where \(T(t) = $func ,\hspace{0.2in} T(t+1) = T(t) \). 
$PAR
\(Y(s)= \) \{ans_rule(40) \}.
$BR Graph of  \( T(t) \) (a triangular wave function):
$BR $g
END_TEXT

Context("Numeric")->variables->are('s'=>"Real");

$ans = Compute("$ans5");
$ans->{limits} = [1,2];

ANS($ans->cmp());

ENDDOCUMENT() ;        # This should be the last executable line in the problem.