##DESCRIPTION
##KEYWORDS('differential', 'equation', 'separable', 'solution', 'initial', 'value')
## tcao tagged and PAID on 3-22-2004

## DBsubject('Calculus')
## DBchapter('Differential Equations')
## DBsection('Separable Equations')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('9.3')
## Problem1('11')
##
##ENDDESCRIPTION

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

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGgraphmacros.pl",
"PGmatrixmacros.pl", 
"PGnumericalmacros.pl",
"PGauxiliaryFunctions.pl",
"PGmorematrixmacros.pl"
);

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

$a = random(2,9,1); 
$t0 = 0;
$x0 = random(1,9,1); 

$c = $x0 * $x0 - 2* $a * $t0;

BEGIN_TEXT

Solve the separable differential equation 
\[ \frac{dx}{dt} = \frac{$a}{x} ,\]
and find the particular solution satisfying the initial condition 
\[ x($t0) = $x0. \]
$BR
\( x(t) = \) \{ ans_rule(40) \}. 

END_TEXT

ANS(fun_cmp("sqrt(2*$a*t+$c)", vars => 't'));

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

##KEYWORDS()
## tcao tagged and PAID on 3-22-2004

## DBsubject('Calculus')
## DBchapter('Differential Equations')
## DBsection('Separable Equations')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('9.3')
## Problem1('9 10')
## DESCRIPTION
##  ## Differential equations
##   normalseparable.pg  must be done  previously.
##  dy/dx = ($P + $Q,x + $R,y + $S,xy) )
##   URL:http://webhost.math.rochester.edu/mth163lib/discuss/msgReader$391
## 
## ENDDESCRIPTION
DOCUMENT() ;  # the first executable line

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl",
"PGdiffeqmacros.pl"
) ;

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

#Put constants and parameters here.

$a = random(1,7,1) ;
$b = random(2,9,1) ;
$c = random(1,7,1) ;
$d = random(2,9,1) ;
$P = $a * $c ;
$Q = $b * $c ;
$R = $a * $d ;
$S = $b * $d ;

BEGIN_TEXT

The differential equation 
\[ \frac{dy}{dx} = $P + $Q \,x + $R \,y + $S \,xy \] 
has an implicit  general solution of the form \(F(x,y) = K. \) $BR $BR 
In fact, because the differential equation is  separable,
we can define the solution curve implicitly by a function in the form 
\[F(x,y) = G(x) + H(y) =K. \]
$BR $BR 
Find such a solution and then give the related functions requested. $BR
 \( F(x,y) = G(x) + H(y) = \)  \{ans_box(3,60) \}. 
  
END_TEXT
     
$ansF = "x*(2*$a + $b*x)/(2*$a +$b )" ;

$ansG = "(- 2 /($d*$b + 2*$a*$d) )* log($c + $d * y)" ;
$ans = "x*(2*$a + $b*x)/(2*$a +$b ) + (- 2 /($d*$b + 2*$a*$d) )* log($c + $d * y)";


ANS fun_cmp("k($ans) +c", var=>['x','y'], params=>['k','c'] ) ;


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

##################################################
my $XML_INFORMATION = <<'END_OF_XML_TRAILER_INFO';
<?xml version="1.0"?>
<metaPGdata>
        <author>Webwork Team</author>
        <course>MTH163</course>
        <description>Differential equations
 normalseparable.pg  must be done  previously.
dy/dx = ($P + $Q,x + $R,y + $S,xy) )</description>
        <fullPath>setDESep/6.pg</fullPath>
        <institution>University of Rochester</institution>
        <keywords>Differential Equation,separable,
first order nonlinear,
normalized,general solution</keywords>
        <libraryPath>setDESep/6.pg</libraryPath>
        <libraryURL>http://webhost.math.rochester.edu/mth163lib/discuss/msgReader$391</libraryURL>
        <modified><dateTime.iso8601>20000713T09:36:43</dateTime.iso8601></modified>
        <msgNum>391</msgNum>
        <pgProblem>true</pgProblem>
        <preface></preface>
        <problemVariants></problemVariants>
        <probNum></probNum>
        <psvn></psvn>
        <revisedVersions></revisedVersions>
        <setName>DESep</setName>
        <titleRoot>6</titleRoot>
        </metaPGdata>

END_OF_XML_TRAILER_INFO
##################################################
________________________________________________________________________________

##DESCRIPTION
##  Separable Differential Equation
##ENDDESCRIPTION

##KEYWORDS('Differential Equation', 'Separable')
## tcao tagged and PAID on 3-22-2004

## DBsubject('Calculus')
## DBchapter('Differential Equations')
## DBsection('Separable Equations')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('9.3')
## Problem1('4')

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

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

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

$a = random(2,10,1);
$b = random(2,6,1);
$n = random(2,17,1);
$n1 = $n +1;
$a1 =$a*$n1;

BEGIN_TEXT

Find \(f(x) \) if  \(y = f(x) \)  satisfies 
\[  \frac{dy}{dx} = $a1 yx^{$n} \]
and the \( y \)-intercept of the curve  \(y = f(x) \) is \( $b \).
$BR
\(f(x) =\) \{ ans_rule(40) \}. $BR

END_TEXT

$ans = "$b * exp($a * (x^$n1 ) )";

ANS(fun_cmp($ans));
ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

##DESCRIPTION
##  Separable Differential Equation
##ENDDESCRIPTION

##KEYWORDS('Differential Equation', 'Separable')
## tcao tagged and PAID on 3-22-2004

## DBsubject('Calculus')
## DBchapter('Differential Equations')
## DBsection('Separable Equations')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('9.3')
## Problem1('12')

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

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGanswermacros.pl",
);

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

$k = non_zero_random(-10,10);
$a = random(2,10);
$b = random(2,6);
$d = $a*2/$b;
$p = -$d + $k*$k;

BEGIN_TEXT
Solve the separable differential equation 
\[  $a x - $b y \sqrt{x^2 + 1} \frac{dy}{dx} = 0. \]
Subject to the initial condition: \( y(0) = $k \).
$BR
\(y =\) \{ ans_rule(50) \}.
END_TEXT

if ($k >=0) {
$ans = "sqrt($d sqrt(x^2 +1) + $p)";
} else {
$ans = "-sqrt($d sqrt(x^2 +1) + $p)";
}


ANS(fun_cmp($ans));
ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

##DESCRIPTION
##KEYWORDS('differential equations', 'separable')
## tcao tagged and PAID on 3-22-2004

## DBsubject('Calculus')
## DBchapter('Differential Equations')
## DBsection('Separable Equations')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('9.3')
## Problem1('6')
##Authored by Zig Fiedorowicz 2/4/2000
##ENDDESCRIPTION

DOCUMENT();

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);
$showPartialCorrectAnswers = 1;

$a = random(2,7,1);
$a1 = $a+1;
$a1i = 1/$a1;

TEXT(beginproblem());

BEGIN_TEXT
Find the solution of the differential equation
\[\left(\ln(y)\right)^$a\frac{dy}{dx} = x^$a y\]
which satisfies the initial condition \(y(1)=e^2\).
$BR
\(y=\) \{ ans_rule(30)\}.
END_TEXT

ANS(fun_cmp("e^((x^$a1+2^$a1-1)^$a1i)", vars=>"x"));

ENDDOCUMENT();
________________________________________________________________________________

##DESCRIPTION
##KEYWORDS('differential equations', 'separable')
## tcao tagged and PAID on 3-22-2004

## DBsubject('Calculus')
## DBchapter('Differential Equations')
## DBsection('Separable Equations')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('9.3')
## Problem1('10')
##Ellis & Gulick section 6.8
##Authored by Zig Fiedorowicz 4/17/2000
##ENDDESCRIPTION

DOCUMENT();

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

$showPartialCorrectAnswers = 1;

$aa = random(2,8,1);
$bb = 1+$aa;

TEXT(beginproblem());
BEGIN_TEXT
Find the particular solution of the differential equation
\[\frac{x^2}{y^2-$aa}\frac{dy}{dx} = \frac{1}{2y}\]
satisfying the initial condition \(y(1)=\sqrt{$bb}\).
$BR
Answer: \(y=\)  \{ ans_rule(45)\}.
$BR
Your answer should be a function of \(x\).
END_TEXT

##set $PG_environment{'textbook'} in webworkCourse.ph
if (defined($textbook)) {
   if ($textbook eq "EllisGulick5") {
BEGIN_TEXT
$PAR
Note this is similar to problem 2 in Section 6.8 of the text.
END_TEXT
}
}

ANS(fun_cmp("sqrt(e^(1-1/x)+$aa)", vars=>"x"));

ENDDOCUMENT();
________________________________________________________________________________

##DESCRIPTION
##KEYWORDS('differential equations', 'separable')
## tcao tagged and PAID on 3-22-2004

## DBsubject('Calculus')
## DBchapter('Differential Equations')
## DBsection('Separable Equations')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('9.3')
## Problem1('14')
##Ellis & Gulick section 6.8
##Authored by Zig Fiedorowicz 4/17/2000
##ENDDESCRIPTION

DOCUMENT();

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

$showPartialCorrectAnswers = 1;

$aa = random(2,8,1);
$bb = random(3,9,1);
if ($aa==$bb){$aa++;}
$b3 = $bb**3;

TEXT(beginproblem());
BEGIN_TEXT
Find the particular solution of the differential equation
\[\frac{dy}{dx} = (x-$aa)e^{-2y}\]
satisfying the initial condition \(y($aa)=\ln($aa)\).
$BR
Answer: \(y=\)  \{ ans_rule(45)\}.
$BR
Your answer should be a function of \(x\).
END_TEXT

##set $PG_environment{'textbook'} in webworkCourse.ph
if (defined($textbook)) {
   if ($textbook eq "EllisGulick5") {
BEGIN_TEXT
$PAR
Note that this is similar to problem 7 in Section 6.8 of the
text.
END_TEXT
}
}

ANS(fun_cmp("(1/2)*ln((x-$aa)^2+$aa^2)", vars=>"x"));

ENDDOCUMENT();
________________________________________________________________________________

##DESCRIPTION
##KEYWORDS('differential', 'equation')
## tcao tagged and PAID on 3-22-2004

## DBsubject('Calculus')
## DBchapter('Differential Equations')
## DBsection('Separable Equations')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('9.3')
## Problem1('28')
##
##ENDDESCRIPTION

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

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGgraphmacros.pl",
"PGmatrixmacros.pl", 
"PGnumericalmacros.pl",
"PGauxiliaryFunctions.pl",
"PGmorematrixmacros.pl"
);

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

$a = random(2,9,1);
@slope_word = ('twice', 'three times', 'four times', 'five times');
@slope_num = (2,3,4,5);
$tag = random(0,3,1);

BEGIN_TEXT

A curve passes through the point \( (0,$a) \) and has the property that the slope of the curve at every point \(P\) is
$slope_word[$tag] the \(y\)-coordinate of \(P\). What is the equation of the curve? 
$BR 
\(y(x)=\) \{ans_rule(10)\} 

END_TEXT

ANS(fun_cmp("$a * e^($slope_num[$tag] * x)"));

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

##DESCRIPTION
##KEYWORDS('differential', 'equation')
## tcao tagged and PAID on 3-22-2004

## DBsubject('Calculus')
## DBchapter('Differential Equations')
## DBsection('Separable Equations')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('9.3')
## Problem1('37')
##
##ENDDESCRIPTION

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

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGgraphmacros.pl",
"PGmatrixmacros.pl", 
"PGnumericalmacros.pl",
"PGauxiliaryFunctions.pl",
"PGmorematrixmacros.pl"
);

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

@name = (Jim, David, Mark, Bill, Andy, John, Peter, Bob); 
$tag1 = random(0,5,1);
$tag2 = random($tag1+1,7,1);

$a = random(11,12,1);
$b = random(13,15,1);

$c = random(10,$a-1,1);
$d = random($b+1,16); 


BEGIN_TEXT

Let \( P(t) \) be the performance level of someone learning a skill as a function of the training time \(t\).
The derivative \( \displaystyle \frac{dP}{dt} \) represents the rate at which performance improves. If \(M\)
is the maximum level of performance of which the learner is capable, then a model for learning is given by the
differential equation 
\[ \frac{dP}{dt} = k(M-P(t)) \] 
where \(k\) is a positive constant. 
$BR
Two new workers, $name[$tag1] and $name[$tag2], were hired for an assembly line. 
$name[$tag1] could process \($a\) units per minute after one hour and \($b\) units per minute after two
hours. $name[$tag2] could process \($c\) units per minute after one hour and \($d\) units per minute after two
hours. Using the above model and assuming that \(P(0)=0\), estimate the maximum number of units per minute
that each worker is capable of processing. 
$BR
$name[$tag1]: \{ ans_rule(20) \}, 
$BR
$name[$tag2]: \{ ans_rule(20) \}. 

END_TEXT

$max1 = $a/(2 - $b/$a);
$max2 = $c/(2 - $d/$c);

ANS(num_cmp($max1));
ANS(num_cmp($max2));

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

##KEYWORDS('Differential Equation', 'Mixing problem')
## tcao tagged and PAID on 3-22-2004

## DBsubject('Calculus')
## DBchapter('Differential Equations')
## DBsection('Separable Equations')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('9.3')
## Problem1('34')
## DESCRIPTION 
##  ## Differential equations
##   URL:http://webhost.math.rochester.edu/mth163lib/discuss/msgReader$437
##  
## ENDDESCRIPTION 
DOCUMENT() ;        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
) ;

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

$e = exp(1) ;
$a = random(1000,3000,20) ;
$b = random(.01,.09,.01) ;
$c = random(2,9,1) ;
$d = random(30,90,3) ;
$a1 = $a*$b ;
$k = $c/$a ;

BEGIN_TEXT
A tank contains \($a\) L of pure water. A solution 
that contains \($b\) kg of sugar per liter enters 
tank at the rate \($c\) L/min The solution is mixed and drains
from the tank at the same rate. $BR $BR 
(a) How much sugar is in the tank at the beginning.$BR
\(y(0) = \) \{ans_rule(5) \} (include units) $BR $BR 
(b) Find the amount of sugar (in kg) after \(t\)  minutes. $BR
\(y(t) =\) \{ ans_rule(40) \}  (function of \(t\)) $BR $BR 
(b) Find the amout of the sugar after \($d\) minutes. $BR
\( y($d)= \) \{ans_rule(40) \} (include units) 

END_TEXT

$ans0 = $a-$a ;
$ans = "$a1*(1-$e^(-$k*t))" ;
$ans1=  $a1*(1-$e**(-$k*$d)) ;

ANS(num_cmp($ans0, unit => 'kg')) ;
ANS(fun_cmp($ans, vars=>"t")) ;
ANS(num_cmp($ans1, unit => 'kg')) ;
ENDDOCUMENT() ;        # This should be the last executable line in the problem.

##################################################
my $XML_INFORMATION = <<'END_OF_XML_TRAILER_INFO';
<?xml version="1.0"?>
<metaPGdata>
	<author>Webwork Team</author>
	<course>MTH163</course>
	<description>Differential equations</description>
	<fullPath>setDiffEQ3Mixing/ns7_4_31.pg</fullPath>
	<institution>University of Rochester</institution>
	<keywords>Differential Equation,Mixing problem</keywords>
	<libraryPath>setDiffEQ3Mixing/ns7_4_31.pg</libraryPath>
	<libraryURL>http://webhost.math.rochester.edu/mth163lib/discuss/msgReader$437</libraryURL>
	<modified><dateTime.iso8601>20000718T14:49:50</dateTime.iso8601></modified>
	<msgNum>437</msgNum>
	<pgProblem>true</pgProblem>
	<preface></preface>
	<problemVariants></problemVariants>
	<probNum></probNum>
	<psvn></psvn>
	<revisedVersions></revisedVersions>
	<setName>DiffEQ3Mixing</setName>
	<titleRoot>ns7_4_31</titleRoot>
	</metaPGdata>

END_OF_XML_TRAILER_INFO
##################################################