##
## Tagging and Description
##------------------------
##
## DBsubject(Differential equations)
## DBchapter(Partial differential equations)
## DBsection()
##    -- recommended section: Solutions and boundary conditions
## Level(1)
## KEYWORDS('partial differential equations','boundary condition')
## TitleText1(n/a)
## EditionText1(n/a)
## AuthorText1(n/a)
## Section1(n/a)
## Problem1(n/a)
## Author(Kenneth Luther)
## Institution(Valparaiso University)
## Language(en)
##
## NOTE: This problem is from a course that uses the text Fourier Series and
## Boundary Value Problems, 8th Ed, by Brown & Churchill, however this particular
## problem is not from that text.   The majority of problems in the Valparaiso
## PDE problem library will come from that text, though.   
##
## Initialization
##---------------
##
DOCUMENT();
loadMacros(
"PGstandard.pl",
"MathObjects.pl",
"PGcourse.pl",
);

##
## Problem Set Up
##---------------
##

Context("Numeric");

$answer = Compute("-3");

##
## Problem Display / Text
##-----------------------
##
TEXT(beginproblem());
Context()-> texStrings;
BEGIN_TEXT

Suppose the function \( u(x,t) = C e^{-t/2} \sin(\pi x)  \) is the general solution to some PDE.   What is the value of \( C \) for which 
this general solution will satisfy the initial condition \( \displaystyle u \left( \frac{1}{2} , 0 \right) = -3 \, \) ?  
$PAR
\( C = \) \{ ans_rule(5) \}

END_TEXT
Context()->normalStrings;

##
## Answer Evaluation and Solution
##-------------------------------
##
ANS( $answer -> cmp() );

Context()->texStrings;
SOLUTION(EV3(<<'END_SOLUTION'));
$PAR
Solution:
$PAR
The general solution specifically when \( \displaystyle x = \frac{1}{2}, t = 0 \) has the form
$BR
\( \displaystyle u \left( \frac{1}{2}, 0 \right) = C e^{0} \sin \left( \pi \cdot \frac{1}{2} \right)  = C (1) = C \)
$BR
If we require \( \displaystyle u \left( \frac{1}{2}, 0 \right) = -3 \), then we have \( C = -3 \).   

END_SOLUTION;
Context()->normalStrings;

ENDDOCUMENT();
