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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 26 " Math 217 Fall 2001 Exam \+
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{TEXT -1 10 "Exam Data\n" }}{PARA 0 "" 0 "" {TEXT 611 12 "Median Score
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00:  31\n65 -  79 :  33\n50 - 64  :  36\n40 - 49  :  22\n  0 - 39  :  \+
24 \n\nFollowing each correct letter choice given below, the percentag
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2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 4 "a)  " }{XPPEDIT 19 1 "y^
2;" "6#*$%\"yG\"\"#" }{TEXT -1 12 "        b)  " }{XPPEDIT 19 1 "x^2;
" "6#*$%\"xG\"\"#" }{TEXT -1 14 "         c)   " }{XPPEDIT 19 1 "x^2-y
^2;" "6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF&!\"\"" }{TEXT -1 11 "        d) \+
" }{XPPEDIT 19 1 "x^2+y^2;" "6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF&F'" }
{TEXT -1 11 "       e)  " }{XPPEDIT 19 1 "abs(x+y);" "6#-%$absG6#,&%\"
xG\"\"\"%\"yGF(" }{TEXT -1 5 "\n\nf) " }{XPPEDIT 19 1 "y^3;" "6#*$%\"y
G\"\"$" }{TEXT -1 13 "         g)  " }{XPPEDIT 19 1 "xy;" "6#%#xyG" }
{TEXT -1 14 "          h)  " }{XPPEDIT 19 1 "1-y;" "6#,&\"\"\"F$%\"yG!
\"\"" }{TEXT -1 16 "             i) " }{XPPEDIT 19 1 "1+x;" "6#,&\"\"
\"F$%\"xGF$" }{TEXT -1 15 "            j) " }{XPPEDIT 19 1 "x^3" "6#*$
%\"xG\"\"$" }{TEXT -1 3 "   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 569 9 "Solution:" }
{TEXT 570 13 "   (i)  (79%)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 340 "The graph has arrows that point down.  Therefo
re answers (a), (b), (d), and (e)  may be eliminated. The slopes of th
e arrows do not depend on  y. Therefore answers (c), (f), (g), and (h)
 may also be eliminated.  It comes down to only (i) and (j).  The last
 one is ruled out since for  -1 < x < 0 the arrows have positive slope
 and x^3 < 0. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 477 18 "2. Suppose that
   " }{XPPEDIT 481 1 "M;" "6#%\"MG" }{TEXT 480 8 "   and  " }{XPPEDIT 
260 1 "N;" "6#%\"NG" }{TEXT 482 108 "  are continuous functions of one
 variable.  Which of the following statements best describes the equat
ion  " }{XPPEDIT 478 1 "dy/dx = (M(x)-y)/(x+N(y));" "6#/*&%#dyG\"\"\"%
#dxG!\"\"*&,&-%\"MG6#%\"xGF&%\"yGF(F&,&F.F&-%\"NG6#F/F&F(" }{TEXT 479 
21 " ?  (In the answers, " }{XPPEDIT 19 1 "C;" "6#%\"CG" }{TEXT 483 
16 " is a constant.)" }}{PARA 0 "" 0 "" {TEXT -1 30 "a)  The equation \+
is separable." }}{PARA 0 "" 0 "" {TEXT -1 32 "b)  The equation is homo
geneous." }}{PARA 0 "" 0 "" {TEXT -1 27 "c)  The equation is linear." 
}}{PARA 0 "" 0 "" {TEXT -1 30 "d)  The equation is Bernoulli." }}
{PARA 0 "" 0 "" {TEXT -1 39 "e) The solution is given explicitly by " 
}{XPPEDIT 19 1 "y(x) = (Int(M(x),x)-xy)/(x*N(y)+x^2/2)+C;" "6#/-%\"yG6
#%\"xG,&*&,&-%$IntG6$-%\"MG6#F'F'\"\"\"%#xyG!\"\"F1,&*&F'F1-%\"NG6#F%F
1F1*&F'\"\"#F:F3F1F3F1%\"CGF1" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 18 "f) The function   " }{XPPEDIT 18 0 "mu(x,y) = -x+y;" "6#/
-%#muG6$%\"xG%\"yG,&F'!\"\"F(\"\"\"" }{TEXT -1 28 "   is an integratin
g factor." }}{PARA 0 "" 0 "" {TEXT -1 42 "g)  The solution is given im
plicitly by   " }{XPPEDIT 19 1 "2*xy+Int(N(y),y) = Int(M(x),x)+C;" "6#
/,&*&\"\"#\"\"\"%#xyGF'F'-%$IntG6$-%\"NG6#%\"yGF/F',&-F*6$-%\"MG6#%\"x
GF6F'%\"CGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 40 "h)  The s
olution is given implicitly by " }{XPPEDIT 19 1 "C = x*y-Int(M(x),x)+I
nt(N(y),y);" "6#/%\"CG,(*&%\"xG\"\"\"%\"yGF(F(-%$IntG6$-%\"MG6#F'F'!\"
\"-F+6$-%\"NG6#F)F)F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 40 "
i)  The solution is given implicitly by " }{XPPEDIT 19 1 "Int(N(y),y) \+
= Int(M(x),x)+C;" "6#/-%$IntG6$-%\"NG6#%\"yGF*,&-F%6$-%\"MG6#%\"xGF1\"
\"\"%\"CGF2" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 33 "j)  The e
quation has no solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 571 9 "Solution:" }{TEXT 
572 14 "   (h)   (26%)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 34 "The equation may be written as    " }{XPPEDIT 19 1 "
(y-M(x))*dx+(x+N(y))*dy = 0;" "6#/,&*&,&%\"yG\"\"\"-%\"MG6#%\"xG!\"\"F
(%#dxGF(F(*&,&F,F(-%\"NG6#F'F(F(%#dyGF(F(\"\"!" }{TEXT -1 10 ".  Since
  " }{XPPEDIT 19 1 "diff(y-M(x),y) = diff(x+N(y),x);" "6#/-%%diffG6$,&
%\"yG\"\"\"-%\"MG6#%\"xG!\"\"F(-F%6$,&F-F)-%\"NG6#F(F)F-" }{TEXT -1 
60 "   (namely, both sides are 1) the equation is exact.  Let   " }
{XPPEDIT 19 1 "F(x,y) = Int(y-M(x),x)+g(y);" "6#/-%\"FG6$%\"xG%\"yG,&-
%$IntG6$,&F(\"\"\"-%\"MG6#F'!\"\"F'F.-%\"gG6#F(F." }{TEXT -1 7 "  or  \+
 " }{XPPEDIT 19 1 "F(x,y) = xy-Int(M(x),x)+g(y);" "6#/-%\"FG6$%\"xG%\"
yG,(%#xyG\"\"\"-%$IntG6$-%\"MG6#F'F'!\"\"-%\"gG6#F(F+" }{TEXT -1 10 ".
   Then  " }{XPPEDIT 19 1 "diff(F(x,y),x) = y-M(x);" "6#/-%%diffG6$-%
\"FG6$%\"xG%\"yGF*,&F+\"\"\"-%\"MG6#F*!\"\"" }{TEXT -1 8 "  and   " }
{XPPEDIT 19 1 "diff(F(x,y),y) = x+D(g)(y);" "6#/-%%diffG6$-%\"FG6$%\"x
G%\"yGF+,&F*\"\"\"--%\"DG6#%\"gG6#F+F-" }{TEXT -1 9 ".   Set  " }
{XPPEDIT 19 1 "x+D(g)(y) = x+N(y);" "6#/,&%\"xG\"\"\"--%\"DG6#%\"gG6#%
\"yGF&,&F%F&-%\"NG6#F-F&" }{TEXT -1 16 "  and solve for " }{XPPEDIT 
19 1 "g;" "6#%\"gG" }{TEXT -1 4 " :  " }{XPPEDIT 19 1 "g(y) = Int(N(y)
,y);" "6#/-%\"gG6#%\"yG-%$IntG6$-%\"NG6#F'F'" }{TEXT -1 14 ".  Therefo
re  " }{XPPEDIT 19 1 "F(x,y) = x*y-Int(M(x),x)+Int(N(y),y);" "6#/-%\"F
G6$%\"xG%\"yG,(*&F'\"\"\"F(F+F+-%$IntG6$-%\"MG6#F'F'!\"\"-F-6$-%\"NG6#
F(F(F+" }{TEXT -1 8 ".  and  " }{XPPEDIT 19 1 "F(x,y) = C;" "6#/-%\"FG
6$%\"xG%\"yG%\"CG" }{TEXT -1 18 "  is the solution." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 361 16 "3. Suppose that
 " }{XPPEDIT 362 1 "y(x);" "6#-%\"yG6#%\"xG" }{TEXT 363 45 " is the so
lution of the initial value problem" }}{PARA 3 "" 0 "" {TEXT 372 1 " \+
" }{XPPEDIT 364 1 "x*diff(y(x),x) = -3*y(x)^2,y(1) = 10;" "6$/*&%\"xG
\"\"\"-%%diffG6$-%\"yG6#F%F%F&,$*&\"\"$F&*$-F+6#F%\"\"#F&!\"\"/-F+6#F&
\"#5" }{TEXT 365 11 ".    Then  " }{XPPEDIT 366 1 "y(exp(1));" "6#-%\"
yG6#-%$expG6#\"\"\"" }{TEXT 367 15 "  is equal to:\n" }}{PARA 3 "" 0 "
" {TEXT 487 4 "a)  " }{XPPEDIT 489 0 "1/3;" "6#*&\"\"\"F$\"\"$!\"\"" }
{TEXT 488 14 "           b) " }{XPPEDIT 492 0 "2/3;" "6#*&\"\"#\"\"\"
\"\"$!\"\"" }{TEXT 490 31 "         c) 10 + 3e        d)  " }{XPPEDIT 
493 0 "11/10;" "6#*&\"#6\"\"\"\"#5!\"\"" }{TEXT 491 11 "        e) " }
{XPPEDIT 495 0 "1/13;" "6#*&\"\"\"F$\"#8!\"\"" }{TEXT 494 3 "   " }}
{PARA 3 "" 0 "" {TEXT 360 3 "f) " }{XPPEDIT 368 1 "13;" "6#\"#8" }
{TEXT 336 15 "           g)  " }{XPPEDIT 369 1 "10/31;" "6#*&\"#5\"\"
\"\"#J!\"\"" }{TEXT 337 10 "       h) " }{XPPEDIT 370 1 "10/3;" "6#*&
\"#5\"\"\"\"\"$!\"\"" }{TEXT 338 17 "              i) " }{XPPEDIT 371 
1 "5/2;" "6#*&\"\"&\"\"\"\"\"#!\"\"" }{TEXT 339 16 "           j)   " 
}{XPPEDIT 529 1 "15/2;" "6#*&\"#:\"\"\"\"\"#!\"\"" }}{PARA 3 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
573 9 "Solution:" }{TEXT 574 14 "   (g)   (65%)" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 54 "dsolve(\{x*diff(y(x),x) = -3*y(x)^2, y(1) = 10\}, \+
y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&\"\"\"F),&-%
#lnGF&\"\"$#F)\"#5F)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
20 "subs(x = exp(1), %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#-
%$expG6#\"\"\"*&F*F*,&-%#lnGF&\"\"$#F*\"#5F*!\"\"" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%\"yG6#-%$expG6#\"\"\"#\"#5\"#J" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 
373 12 " 4. Find    " }{XPPEDIT 374 1 "y(Pi/4);" "6#-%\"yG6#*&%#PiG\"
\"\"\"\"%!\"\"" }{TEXT 375 8 "    if  " }{XPPEDIT 376 1 "dy/dx = (x+y)
^2,y(0) = 0;" "6$/*&%#dyG\"\"\"%#dxG!\"\"*$,&%\"xGF&%\"yGF&\"\"#/-F,6#
\"\"!F1" }{TEXT 377 1 "." }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }{TEXT 456 4 "a)  " }{XPPEDIT 259 0 "0;" "6#\"\"!" }{TEXT 260 21 
"                b)   " }{XPPEDIT 261 0 "1;" "6#\"\"\"" }{TEXT 262 18 
"             c)   " }{XPPEDIT 263 0 "Pi/4;" "6#*&%#PiG\"\"\"\"\"%!\"
\"" }{TEXT 264 19 "              d)   " }{XPPEDIT 265 0 "Pi/2;" "6#*&%
#PiG\"\"\"\"\"#!\"\"" }{TEXT 266 19 "              e)   " }{XPPEDIT 
267 0 "Pi;" "6#%#PiG" }{TEXT 268 9 "        \n" }{TEXT 340 5 "f)   " }
{XPPEDIT 269 0 "Pi+1;" "6#,&%#PiG\"\"\"F%F%" }{TEXT 270 12 "        g)
  " }{XPPEDIT 271 0 "Pi/2+1;" "6#,&*&%#PiG\"\"\"\"\"#!\"\"F&F&F&" }
{TEXT 272 12 "        h)  " }{XPPEDIT 273 0 "Pi/4-1;" "6#,&*&%#PiG\"\"
\"\"\"%!\"\"F&F&F(" }{TEXT 274 11 "       i)  " }{XPPEDIT 275 0 "Pi/4+
1;" "6#,&*&%#PiG\"\"\"\"\"%!\"\"F&F&F&" }{TEXT 276 13 "         j)  " 
}{XPPEDIT 277 0 "1-Pi/4;" "6#,&\"\"\"F$*&%#PiGF$\"\"%!\"\"F(" }}{PARA 
3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 575 9 "Solution:" }
{TEXT 576 14 "   (j)   (38%)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "ds
olve(\{diff(y(x),x)= (x+y(x))^2, y(0) = 0\}, y(x));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&-%$tanG6#,$F'!\"\"F-F'F-" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "subs(x = Pi/4, %);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%\"yG6#,$%#PiG#\"\"\"\"\"%,&-%$tanGF&F**&#F*F+F*
F(F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#,$%#PiG#\"\"\"\"\"%,&F*F**&
#F*F+F*F(F*!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 43 "To do this by hand make the substitution   " }{XPPEDIT 
19 1 "u = x+y;" "6#/%\"uG,&%\"xG\"\"\"%\"yGF'" }{TEXT -1 41 "   in the
 original differential equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 457 2 " 5" }
{TEXT -1 2 ". " }{TEXT 357 87 "What is a sensible first step towards f
inding a solution to the differential equation \n" }{XPPEDIT 341 1 "dy
/dx = f(x,y);" "6#/*&%#dyG\"\"\"%#dxG!\"\"-%\"fG6$%\"xG%\"yG" }{TEXT 
342 6 "  if  " }{XPPEDIT 532 1 "f(t*x,t*y) = f(x,y);" "6#/-%\"fG6$*&%
\"tG\"\"\"%\"xGF)*&F(F)%\"yGF)-F%6$F*F," }{TEXT 531 14 "   for every  \+
" }{XPPEDIT 533 1 "t;" "6#%\"tG" }{TEXT 530 2 " ?" }}{PARA 3 "" 0 "" 
{TEXT 343 32 "a) Make the change of variable  " }{XPPEDIT 359 0 "v(x) \+
= x+y;" "6#/-%\"vG6#%\"xG,&F'\"\"\"%\"yGF)" }{TEXT 358 34 ".\nb) Make \+
the change of variable  " }{XPPEDIT 344 1 "v(x) = x*y;" "6#/-%\"vG6#%
\"xG*&F'\"\"\"%\"yGF)" }{TEXT 345 34 ".\nc) Make the change of variabl
e  " }{XPPEDIT 346 1 "v(x) = y(x)^2;" "6#/-%\"vG6#%\"xG*$-%\"yG6#F'\"
\"#" }{TEXT 347 1 "." }}{PARA 3 "" 0 "" {TEXT 348 32 "d) Make the chan
ge of variable  " }{XPPEDIT 349 1 "v(x) = y(x)^t;" "6#/-%\"vG6#%\"xG)-
%\"yG6#F'%\"tG" }{TEXT 350 1 "." }}{PARA 3 "" 0 "" {TEXT 351 32 "e) Ma
ke the change of variable  " }{XPPEDIT 257 1 "v(x) = y/x;" "6#/-%\"vG6
#%\"xG*&%\"yG\"\"\"F'!\"\"" }{TEXT 534 1 "." }}{PARA 3 "" 0 "" {TEXT 
352 42 "f) Multiply both sides of the equation by " }{XPPEDIT 538 1 "e
xp(Int(f(x,y),x));" "6#-%$expG6#-%$IntG6$-%\"fG6$%\"xG%\"yGF," }{TEXT 
535 4 "   ." }}{PARA 3 "" 0 "" {TEXT 353 42 "g) Multiply both sides of
 the equation by " }{XPPEDIT 539 1 "exp(Int(f(t*x,t*y),t));" "6#-%$exp
G6#-%$IntG6$-%\"fG6$*&%\"tG\"\"\"%\"xGF.*&F-F.%\"yGF.F-" }{TEXT 536 
35 " .\nh) Make the change of variable  " }{XPPEDIT 540 1 "v = x/t;" "
6#/%\"vG*&%\"xG\"\"\"%\"tG!\"\"" }{TEXT 537 35 " .\ni) Make the change
 of variable  " }{XPPEDIT 541 1 "v = y/t;" "6#/%\"vG*&%\"yG\"\"\"%\"tG
!\"\"" }{TEXT 354 16 ".\nj) Substitute " }{XPPEDIT 355 1 "y(x) = C*(In
t(f(t*x,y),t)+Int(f(x,t*y),t));" "6#/-%\"yG6#%\"xG*&%\"CG\"\"\",&-%$In
tG6$-%\"fG6$*&%\"tGF*F'F*F%F3F*-F-6$-F06$F'*&F3F*F%F*F3F*F*" }{TEXT 
356 17 " and solve for C." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT 577 9 "Solution:" }{TEXT 578 15 "   (e)    (84%)" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 118 "The give
n equation is homogeneous of degree 0. The change of variable in (e) c
onverts the equation to a separable one." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 
-1 1 " " }{TEXT 413 53 "6. Find an integrating factor for the equation
\n      " }{XPPEDIT 414 1 "diff(y(x),x)+2*y(x)/x = 4*x^3;" "6#/,&-%%di
ffG6$-%\"yG6#%\"xGF+\"\"\"*(\"\"#F,-F)6#F+F,F+!\"\"F,*&\"\"%F,*$F+\"\"
$F," }{TEXT 415 69 "\nwhen it is solved using the method for first ord
er linear equations." }}{PARA 0 "" 0 "" {TEXT 416 6 "a)    " }
{XPPEDIT 417 1 "x;" "6#%\"xG" }{TEXT 418 22 "                   b) " }
{XPPEDIT 419 1 "2*x;" "6#*&\"\"#\"\"\"%\"xGF%" }{TEXT 420 13 "        \+
  c) " }{XPPEDIT 421 1 "1/x;" "6#*&\"\"\"F$%\"xG!\"\"" }{TEXT 422 21 "
                  d) " }{XPPEDIT 423 1 "2/x;" "6#*&\"\"#\"\"\"%\"xG!\"
\"" }{TEXT 424 14 "           e) " }{XPPEDIT 425 1 "x^2;" "6#*$%\"xG\"
\"#" }{TEXT 426 7 "\n\nf)   " }{XPPEDIT 427 1 "2*ln(x);" "6#*&\"\"#\"
\"\"-%#lnG6#%\"xGF%" }{TEXT 428 14 "           g) " }{XPPEDIT 429 1 "x
/2;" "6#*&%\"xG\"\"\"\"\"#!\"\"" }{TEXT 430 15 "            h) " }
{XPPEDIT 431 1 "4*x^3;" "6#*&\"\"%\"\"\"*$%\"xG\"\"$F%" }{TEXT 432 17 
"              i) " }{XPPEDIT 433 1 "exp(2/x);" "6#-%$expG6#*&\"\"#\"
\"\"%\"xG!\"\"" }{TEXT 434 11 "        j) " }{XPPEDIT 435 1 "x^4;" "6#
*$%\"xG\"\"%" }{TEXT 436 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 579 9 "Solution:
" }{TEXT 580 14 "   (e)   (91%)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 35 "The required integrating factor is:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "exp(Int(2/x,x)) = exp(int(2/
x,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$expG6#-%$IntG6$,$*&\"\"
\"F,%\"xG!\"\"\"\"#F-*$)F-F/F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " \+
" }{TEXT 437 9 "7. Find  " }{XPPEDIT 438 1 "y(1);" "6#-%\"yG6#\"\"\"" 
}{TEXT 439 4 " if " }{XPPEDIT 440 1 "y(t);" "6#-%\"yG6#%\"tG" }{TEXT 
441 48 " is the solution of the initial value problem\n  " }{XPPEDIT 
442 1 "dy/dt+3*y(t) = 2*exp(-3*t),y(0) = 2;" "6$/,&*&%#dyG\"\"\"%#dtG!
\"\"F'*&\"\"$F'-%\"yG6#%\"tGF'F'*&\"\"#F'-%$expG6#,$*&F+F'F/F'F)F'/-F-
6#\"\"!F1" }{TEXT 443 1 "." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
444 3 "a) " }{XPPEDIT 257 1 "3*e;" "6#*&\"\"$\"\"\"%\"eGF%" }{TEXT 
445 12 "        b)  " }{XPPEDIT 258 1 "3;" "6#\"\"$" }{TEXT 446 12 "  \+
       c) " }{XPPEDIT 260 1 "3/e;" "6#*&\"\"$\"\"\"%\"eG!\"\"" }{TEXT 
447 13 "          d) " }{XPPEDIT 262 1 "3/(e^2)" "6#*&\"\"$\"\"\"*$%\"
eG\"\"#!\"\"" }{TEXT 448 13 "          e) " }{XPPEDIT 264 1 "e^2;" "6#
*$%\"eG\"\"#" }{TEXT 449 4 "\nf) " }{XPPEDIT 266 1 "2*e^2;" "6#*&\"\"#
\"\"\"*$%\"eGF$F%" }{TEXT 450 10 "       g) " }{XPPEDIT 268 1 "3*exp(3
);" "6#*&\"\"$\"\"\"-%$expG6#F$F%" }{TEXT 451 11 "       h)  " }
{XPPEDIT 270 1 "4*exp(3);" "6#*&\"\"%\"\"\"-%$expG6#\"\"$F%" }{TEXT 
452 14 "           i) " }{XPPEDIT 272 1 "4/(e^3);" "6#*&\"\"%\"\"\"*$%
\"eG\"\"$!\"\"" }{TEXT 453 14 "           j) " }{XPPEDIT 274 1 "4/(e^2
)" "6#*&\"\"%\"\"\"*$%\"eG\"\"#!\"\"" }{TEXT 454 2 "  " }}{PARA 0 "" 
0 "" {TEXT 455 0 "" }{TEXT -1 0 "" }{TEXT 333 0 "" }}{PARA 0 "" 0 "" 
{TEXT 581 9 "Solution:" }{TEXT 582 16 "   (i)     (90%)" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "dsolve(\{dif
f(y(t),t)+3*y(t) = 2*exp(-3*t), y(0) = 2\},y(t));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"yG6#%\"tG*&,&F'\"\"#F*\"\"\"F+-%$expG6#,$F'!\"$F+
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(t=1,%);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#\"\"\",$-%$expG6#!\"$\"\"%" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 458 9 "8.  Let  " }
{XPPEDIT 19 1 "f;" "6#%\"fG" }{TEXT 563 111 "  be a continuous functio
n of one variable. Which of the following statements about the differe
ntial equation  " }{XPPEDIT 565 1 "dy/dx = f(y/x);" "6#/*&%#dyG\"\"\"%
#dxG!\"\"-%\"fG6#*&%\"yGF&%\"xGF(" }{TEXT 564 489 "  is/are true?\n(I)
 It can be converted to a linear equation by making a change of variab
le.\n(II) It can be converted to a Bernoulli equation by making a chan
ge of variable.\n(III) It can be converted to a separable equation by \+
making a change of variable. \n\na) None        b) I only        c) II
 only        d) III only        e) I and II only  \nf) II and III only
                g) I and III only                    h) I, II, and III
  \ni) Wrong answer                 j) Bonus wrong answer" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }{TEXT 459 0 "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 583 9 "Solution:" }{TEXT 584 16 "   (d)     (38%)" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "The substitution  " 
}{XPPEDIT 19 1 "v = y/x;" "6#/%\"vG*&%\"yG\"\"\"%\"xG!\"\"" }{TEXT -1 
33 "   leads to a separable equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 460 42 "9.  Consider the differenti
al equation    " }{XPPEDIT 501 1 "dy/dx = 1+y(x)/(x^2);" "6#/*&%#dyG\"
\"\"%#dxG!\"\",&F&F&*&-%\"yG6#%\"xGF&*$F.\"\"#F(F&" }{TEXT 500 42 "  a
nd the following statements about it.  " }}{PARA 3 "" 0 "" {TEXT 461 
206 " (I)        The differential equation is linear.\n(II)       The \+
differential equation is separable.\n(III)      The differential equat
ion is homogeneous. \n(IV)      The differential equation, in the form
    " }{XPPEDIT 258 1 "(x^2+y)*dx-x^2*dy = 0;" "6#/,&*&,&*$%\"xG\"\"#
\"\"\"%\"yGF*F*%#dxGF*F**&F(F)%#dyGF*!\"\"\"\"!" }{TEXT 502 58 ",    i
s exact.\n\nChoose the largest set of true statements." }}{PARA 3 "" 
0 "" {TEXT 499 78 "a)  \{I\}         b)  \{II\}         c)  \{III\}   \+
      d)  \{IV\}         e)  \{I,II\}" }}{PARA 3 "" 0 "" {TEXT 278 77 
"f) \{I,III\}     g) \{I,IV\}      h) \{II,III\}       i) \{II,IV\}   \+
    j) \{III,IV\}\n" }{TEXT 334 0 "" }{TEXT 585 0 "" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 586 9 "Solution:" }{TEXT 587 15 "   (a)    (74%)
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Write
 the equation as   " }{XPPEDIT 19 1 "diff(y(x),x)+(-1/(x^2))*y(x) = 1;
" "6#/,&-%%diffG6$-%\"yG6#%\"xGF+\"\"\"*&,$*&F,F,*$F+\"\"#!\"\"F2F,-F)
6#F+F,F,F," }{TEXT -1 38 "   to see that the equation is linear." }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 462 38 "10. Solve the in
itial value problem   " }{XPPEDIT 463 1 "dy/dx = 1+y(x)/x,y(1) = 4;" "
6$/*&%#dyG\"\"\"%#dxG!\"\",&F&F&*&-%\"yG6#%\"xGF&F.F(F&/-F,6#F&\"\"%" 
}{TEXT 464 14 ".   What is   " }{XPPEDIT 497 1 "y(e);" "6#-%\"yG6#%\"e
G" }{TEXT 496 3 " ? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "
" {TEXT 335 4 "a)  " }{XPPEDIT 279 0 "e/2;" "6#*&%\"eG\"\"\"\"\"#!\"\"
" }{TEXT 280 17 "            b)   " }{XPPEDIT 281 0 "2*e;" "6#*&\"\"#
\"\"\"%\"eGF%" }{TEXT 282 16 "          c)    " }{XPPEDIT 283 0 "3*e;
" "6#*&\"\"$\"\"\"%\"eGF%" }{TEXT 284 15 "          d)   " }{XPPEDIT 
285 0 "3*e/2;" "6#*(\"\"$\"\"\"%\"eGF%\"\"#!\"\"" }{TEXT 286 16 "     \+
      e)   " }{XPPEDIT 287 0 "4*e;" "6#*&\"\"%\"\"\"%\"eGF%" }{TEXT 
288 20 "                    " }}{PARA 3 "" 0 "" {TEXT 498 4 "f)  " }
{XPPEDIT 289 0 "5*e;" "6#*&\"\"&\"\"\"%\"eGF%" }{TEXT 290 15 "        \+
   g)  " }{XPPEDIT 291 0 "5*e/2;" "6#*(\"\"&\"\"\"%\"eGF%\"\"#!\"\"" }
{TEXT 292 16 "           h)   " }{XPPEDIT 293 0 "6*e;" "6#*&\"\"'\"\"
\"%\"eGF%" }{TEXT 294 16 "           i)   " }{XPPEDIT 295 0 "7*e;" "6#
*&\"\"(\"\"\"%\"eGF%" }{TEXT 296 16 "            j)  " }{XPPEDIT 297 
0 "7*e/2;" "6#*(\"\"(\"\"\"%\"eGF%\"\"#!\"\"" }}{PARA 0 "" 0 "" {TEXT 
465 0 "" }}{PARA 0 "" 0 "" {TEXT 588 9 "Solution:" }{TEXT 589 15 "   (
f)    (59%)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "dsolve(\{diff(y(x)
,x) = 1+y(x)/x, y(1) = 4\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/
-%\"yG6#%\"xG*&,&-%#lnGF&\"\"\"\"\"%F,F,F'F," }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 17 "subs(x=exp(1),%);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%\"yG6#-%$expG6#\"\"\"*&,&-%#lnGF&F*\"\"%F*F*F'F*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"yG6#-%$expG6#\"\"\",$F'\"\"&" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 466 89 "11. Which of th
e following is the general  solution \nof  the differential equation: \+
     " }{XPPEDIT 467 0 "dy/dx = -(1+y)/(x+3*y^2);" "6#/*&%#dyG\"\"\"%#
dxG!\"\",$*&,&F&F&%\"yGF&F&,&%\"xGF&*&\"\"$F&*$F,\"\"#F&F&F(F(" }
{TEXT 468 8 "?\n\na)   " }{XPPEDIT 298 1 "x+x*y(x)+y(x)^3 = C;" "6#/,(
%\"xG\"\"\"*&F%F&-%\"yG6#F%F&F&*$-F)6#F%\"\"$F&%\"CG" }{TEXT 299 25 " \+
                   b)   " }{XPPEDIT 300 1 "y(x)^3 = C*x^2+1;" "6#/*$-%
\"yG6#%\"xG\"\"$,&*&%\"CG\"\"\"*$F(\"\"#F-F-F-F-" }{TEXT 301 6 "\nc)  \+
 " }{XPPEDIT 302 1 "x+y(x)+x*y(x) = C;" "6#/,(%\"xG\"\"\"-%\"yG6#F%F&*
&F%F&-F(6#F%F&F&%\"CG" }{TEXT 303 27 "                      d)   " }
{XPPEDIT 469 1 "x*exp(y(x))+y(x)^3 = C;" "6#/,&*&%\"xG\"\"\"-%$expG6#-
%\"yG6#F&F'F'*$-F,6#F&\"\"$F'%\"CG" }{TEXT 378 5 "\ne)  " }{XPPEDIT 
304 1 "y(x) = 3*ln(x)+C;" "6#/-%\"yG6#%\"xG,&*&\"\"$\"\"\"-%#lnG6#F'F+
F+%\"CGF+" }{TEXT 305 33 "                            f)   " }
{XPPEDIT 306 1 "y(x) = e^(C*x^2);" "6#/-%\"yG6#%\"xG)%\"eG*&%\"CG\"\"
\"*$F'\"\"#F," }{TEXT 307 6 "\n\ng)  " }{XPPEDIT 308 1 "x*y(x)+y(x)^3 \+
= C;" "6#/,&*&%\"xG\"\"\"-%\"yG6#F&F'F'*$-F)6#F&\"\"$F'%\"CG" }{TEXT 
309 31 "                           h)  " }{XPPEDIT 310 1 "x*y(x) = C*y
(x)^3;" "6#/*&%\"xG\"\"\"-%\"yG6#F%F&*&%\"CGF&*$-F(6#F%\"\"$F&" }
{TEXT 311 6 "\n\ni)  " }{XPPEDIT 312 1 "x*y(x)+y(x)^3 = -x-x*y(x)+C;" 
"6#/,&*&%\"xG\"\"\"-%\"yG6#F&F'F'*$-F)6#F&\"\"$F',(F&!\"\"*&F&F'-F)6#F
&F'F0%\"CGF'" }{TEXT 313 10 "      j)  " }{XPPEDIT 314 1 "y(x)^2 = x*(
y(x)+C);" "6#/*$-%\"yG6#%\"xG\"\"#*&F(\"\"\",&-F&6#F(F+%\"CGF+F+" }
{TEXT 315 1 "\n" }}{PARA 0 "" 0 "" {TEXT 590 9 "Solution:" }{TEXT 591 
16 "   (a)     (58%)" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 60 "dsolve(diff(y(x),x) = -(1+y(x))/(x+3*y(x)
^2),y(x),implicit);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"xG\"\"\"*
&,&*$)-%\"yG6#F%\"\"$F&!\"\"%$_C1GF&F&,&F&F&F+F&F/F/\"\"!" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 470 54 "12. Which of the followi
ng differential equations is  " }{TEXT 256 3 "not" }{TEXT 471 8 "  exa
ct?" }{TEXT -1 1 "\n" }}{PARA 3 "" 0 "" {TEXT 316 7 "a)     " }
{XPPEDIT 317 1 "2*x*y^3*dx+3*x^2*y^2*dy = 0;" "6#/,&**\"\"#\"\"\"%\"xG
F'%\"yG\"\"$%#dxGF'F'**F*F'*$F(F&F'F)F&%#dyGF'F'\"\"!" }{TEXT 318 8 "
\n\nb)    " }{XPPEDIT 319 1 "cos(x+y)*dx+cos(x+y)*dy = 0;" "6#/,&*&-%$
cosG6#,&%\"xG\"\"\"%\"yGF+F+%#dxGF+F+*&-F'6#,&F*F+F,F+F+%#dyGF+F+\"\"!
" }{TEXT 320 9 "\n\nc)     " }{XPPEDIT 321 1 "e^y*dx+x*e^y*dy = 0;" "6
#/,&*&)%\"eG%\"yG\"\"\"%#dxGF)F)*(%\"xGF))F'F(F)%#dyGF)F)\"\"!" }
{TEXT 322 7 "\n\nd)   " }{XPPEDIT 381 1 "(4*x-y)*dx+(6*y-x)*dy = 0;" "
6#/,&*&,&*&\"\"%\"\"\"%\"xGF)F)%\"yG!\"\"F)%#dxGF)F)*&,&*&\"\"'F)F+F)F
)F*F,F)%#dyGF)F)\"\"!" }{TEXT 379 1 "\n" }{TEXT 380 7 "\ne)    " }
{XPPEDIT 323 1 "(2*x+3*y)*dx+(3*x+2)*dy = 0;" "6#/,&*&,&*&\"\"#\"\"\"%
\"xGF)F)*&\"\"$F)%\"yGF)F)F)%#dxGF)F)*&,&*&F,F)F*F)F)F(F)F)%#dyGF)F)\"
\"!" }{TEXT 324 8 "\n\nf)    " }{XPPEDIT 325 1 "ln(y)*dx+ln(x)*dy = 0;
" "6#/,&*&-%#lnG6#%\"yG\"\"\"%#dxGF*F**&-F'6#%\"xGF*%#dyGF*F*\"\"!" }
{TEXT 326 7 "\n\ng)   " }{XPPEDIT 327 1 "(2*x*y^2+3*x^2)*dx+(2*x^2*y+4
*y^3)*dy = 0;" "6#/,&*&,&*(\"\"#\"\"\"%\"xGF)%\"yGF(F)*&\"\"$F)*$F*F(F
)F)F)%#dxGF)F)*&,&*(F(F)*$F*F(F)F+F)F)*&\"\"%F)*$F+F-F)F)F)%#dyGF)F)\"
\"!" }{TEXT 328 7 "\n\nh)   " }{XPPEDIT 329 1 "(3*x^2+2*y^2)*dx+(4*x*y
+6*y^2)*dy = 0;" "6#/,&*&,&*&\"\"$\"\"\"*$%\"xG\"\"#F)F)*&F,F)*$%\"yGF
,F)F)F)%#dxGF)F)*&,&*(\"\"%F)F+F)F/F)F)*&\"\"'F)*$F/F,F)F)F)%#dyGF)F)
\"\"!" }{TEXT 330 7 "\n\ni)   " }{XPPEDIT 331 1 "Pi*dx+dy/Pi = 0;" "6#
/,&*&%#PiG\"\"\"%#dxGF'F'*&%#dyGF'F&!\"\"F'\"\"!" }{TEXT 332 6 "\nj)  \+
 " }{XPPEDIT 383 1 "ln(x)*dx/(1+x^2)+(1+y^3)*dy/(1+y^2) = 0;" "6#/,&*(
-%#lnG6#%\"xG\"\"\"%#dxGF*,&F*F**$F)\"\"#F*!\"\"F**(,&F*F**$%\"yG\"\"$
F*F*%#dyGF*,&F*F**$F3F.F*F/F*\"\"!" }{TEXT 382 2 ".\n" }}{PARA 0 "" 0 
"" {TEXT 592 9 "Solution:" }{TEXT 593 14 "   (f)   (72%)" }}{PARA 3 "
" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 476 38 " 13. Fin
d the general solution of     " }{XPPEDIT 508 1 "2*x*y*dx+(x^2+1)*dy =
 0;" "6#/,&**\"\"#\"\"\"%\"xGF'%\"yGF'%#dxGF'F'*&,&*$F(F&F'F'F'F'%#dyG
F'F'\"\"!" }{TEXT 507 5 ".\na) " }{XPPEDIT 512 1 "y = C/(x^2);" "6#/%
\"yG*&%\"CG\"\"\"*$%\"xG\"\"#!\"\"" }{TEXT 511 13 "        b)   " }
{XPPEDIT 513 1 "x^2*y+y = C;" "6#/,&*&%\"xG\"\"#%\"yG\"\"\"F)F(F)%\"CG
" }{TEXT 510 11 "       c)  " }{XPPEDIT 515 1 "x*y^2+x = C;" "6#/,&*&%
\"xG\"\"\"*$%\"yG\"\"#F'F'F&F'%\"CG" }{TEXT 514 10 "       d) " }
{XPPEDIT 517 1 "2*x*y+x^2 = C;" "6#/,&*(\"\"#\"\"\"%\"xGF'%\"yGF'F'*$F
(F&F'%\"CG" }{TEXT 516 9 "     \ne) " }{XPPEDIT 519 1 "3*x^2*y+y = C;
" "6#/,&*(\"\"$\"\"\"*$%\"xG\"\"#F'%\"yGF'F'F+F'%\"CG" }{TEXT 518 11 "
       f)  " }{XPPEDIT 521 1 "x^3*y+x*y = C;" "6#/,&*&%\"xG\"\"$%\"yG
\"\"\"F)*&F&F)F(F)F)%\"CG" }{TEXT 520 14 "          g)  " }{XPPEDIT 
523 1 "x^3/3+x+x*y^2 = C;" "6#/,(*&%\"xG\"\"$F'!\"\"\"\"\"F&F)*&F&F)*$
%\"yG\"\"#F)F)%\"CG" }{TEXT 522 9 "    \nh)  " }{XPPEDIT 525 1 "2*x*y/
(x^2+1) = C;" "6#/**\"\"#\"\"\"%\"xGF&%\"yGF&,&*$F'F%F&F&F&!\"\"%\"CG
" }{TEXT 524 14 "          i)  " }{XPPEDIT 527 1 "2*x^2*y+y = C;" "6#/
,&*(\"\"#\"\"\"*$%\"xGF&F'%\"yGF'F'F*F'%\"CG" }{TEXT 526 15 "         \+
  j)  " }{XPPEDIT 528 1 "y = C/(x^3);" "6#/%\"yG*&%\"CG\"\"\"*$%\"xG\"
\"$!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT 594 9 "Solution:" }{TEXT 595 14 "   (b)  \+
 (84%)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "M := 2*x*y:   N := x^2+1
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "F := (x,y) -> int(M,x)
+g(y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGR6$%\"xG%\"yG6\"6$%)op
eratorG%&arrowGF),&-%$intG6$%\"MG9$\"\"\"-%\"gG6#9%F3F)F)F)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "diff(F(x,y),y)= x^2+1;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*$)%\"xG\"\"#\"\"\"F)-%%diffG6$-%\"
gG6#%\"yGF0F),&F%F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "
dsolve(%,g(y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"yG,&F'
\"\"\"%$_C1GF)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "subs(%, F
(x,y)=C);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&)%\"xG\"\"#\"\"\"%\"
yGF)F)F*F)%$_C1GF)%\"CG" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 404 58 "14.
  Which of the following is an integrating factor for  " }{XPPEDIT 
509 1 "(1+x)*dx+x*y*dy = 0;" "6#/,&*&,&\"\"\"F'%\"xGF'F'%#dxGF'F'*(F(F
'%\"yGF'%#dyGF'F'\"\"!" }{TEXT 503 2 " ?" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT 384 4 "a)  " }{XPPEDIT 385 0 "y;" "6#%\"y
G" }{TEXT 386 19 "              b)   " }{XPPEDIT 387 0 "1/y;" "6#*&\"
\"\"F$%\"yG!\"\"" }{TEXT 388 21 "                c)   " }{XPPEDIT 389 
0 "x;" "6#%\"xG" }{TEXT 390 23 "                  d)   " }{XPPEDIT 
391 0 "1/x;" "6#*&\"\"\"F$%\"xG!\"\"" }{TEXT 392 20 "                e
)  " }{XPPEDIT 393 0 "y^2;" "6#*$%\"yG\"\"#" }{TEXT 394 21 "          \+
     \nf)   " }{XPPEDIT 395 0 "1/(y^2);" "6#*&\"\"\"F$*$%\"yG\"\"#!\"
\"" }{TEXT 396 14 "          g)  " }{XPPEDIT 397 0 "x+y;" "6#,&%\"xG\"
\"\"%\"yGF%" }{TEXT 398 17 "            h)   " }{XPPEDIT 399 0 "y/x;" 
"6#*&%\"yG\"\"\"%\"xG!\"\"" }{TEXT 400 21 "                 i)  " }
{XPPEDIT 401 0 "x*y;" "6#*&%\"xG\"\"\"%\"yGF%" }{TEXT 402 21 "        \+
        j)   " }{XPPEDIT 403 0 "x/y;" "6#*&%\"xG\"\"\"%\"yG!\"\"" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 608 9 "Solution:
" }{TEXT 609 15 "   (d)    (60%)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
24 "M := (1+x):    N := x*y:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 38 "exp(int((diff(M,y) - diff(N,x))/N,x));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*&\"\"\"F$%\"xG!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 13 "Verification:" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "diff(M*%,y) - diff(N*%,x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 405 611 "15.  Which \+
of the following statements is/are true?\n(I) A Bernoulli equation can
 be converted to a linear equation by making a change of variable.\n(I
I) A Bernoulli equation can be converted to a homogeneous differential
 equation of degree zero by making a change of variable.\n(III) A Bern
oulli equation can be converted to a separable equation by making a ch
ange of variable. \n\na) None        b) I only        c) II only      \+
  d) III only        e) I and II only  \nf) II and III only           \+
     g) I and III only                     h) I, II, and III  \ni) Wro
ng answer                 j) Bonus wrong answer" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
600 9 "Solution:" }{TEXT 601 16 "   (b)     (45%)" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 
"" {TEXT 406 39 "16. Consider the differential equation " }{XPPEDIT 
543 1 "dy/dx = (x+2*y-4)/(2*x+y-5);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&,(%
\"xGF&*&\"\"#F&%\"yGF&F&\"\"%F(F&,(*&F-F&F+F&F&F.F&\"\"&F(F(" }{TEXT 
542 424 ".   Which of the following changes of variables converts this
 equation to a homogeneous differential equation of degree zero? \n\na
) x = u + 1,  y = v - 2      b) x = u + 1,  y = v + 2        c) x = u \+
- 1,  y = v + 2   \nd) x = u - 1,  y = v - 2      e)  x = u - 2,  y = \+
v + 1         f) x = u - 2,  y = v - 1 \ng) x = u + 2,  y = v - 1     \+
 h)  x = u + 2,  y = v + 1        i) x = u - 2,  y = v + 2\nj) x = u +
 2,  y = v - 2    " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 598 9 "Solution:" }{TEXT 599 14 
"   (h)   (72%)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "
" 0 "" {TEXT 407 290 "17.  A brine (solution of salt in water) initial
ly consists of 50 kg salt dissolved in a 600 liter tank. Pure water en
ters the tank at the rate of 2 liters per minute, is instantly mixed i
nto the brine, and the (diluted) brine exits the tank at the rate of 2
 liters per minute. In how many " }{TEXT 567 5 "hours" }{TEXT 568 53 "
 will there be 25 kg of salt dissolved in the tank? \n" }{TEXT -1 0 "
" }{TEXT 566 87 "\na) 1/3          b) 1/5             c)  3           \+
d)  5              e)  ln(2)  \nf)  " }{XPPEDIT 556 1 "3*ln(2);" "6#*&
\"\"$\"\"\"-%#lnG6#\"\"#F%" }{TEXT 555 7 "    g) " }{XPPEDIT 557 1 "5*
ln(2);" "6#*&\"\"&\"\"\"-%#lnG6#\"\"#F%" }{TEXT 554 11 "       h)  " }
{XPPEDIT 558 1 "ln(2)/3;" "6#*&-%#lnG6#\"\"#\"\"\"\"\"$!\"\"" }{TEXT 
552 9 "    i)   " }{XPPEDIT 559 1 "ln(2)/5;" "6#*&-%#lnG6#\"\"#\"\"\"
\"\"&!\"\"" }{TEXT 553 10 "      j)  " }{XPPEDIT 561 1 "e^(1/3);" "6#)
%\"eG*&\"\"\"F&\"\"$!\"\"" }{TEXT 560 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 606 9 "Solution:" }{TEXT 607 15 "   (g
)    (54%)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 6 "Let   " }{XPPEDIT 18 0 "x(t);" "6#-%\"xG6#%\"tG" }{TEXT -1 30 " \+
  be the mass of salt. Then  " }{XPPEDIT 18 0 "D(x)(t) = -2*x(t)/600;
" "6#/--%\"DG6#%\"xG6#%\"tG,$*(\"\"#\"\"\"-F(6#F*F.\"$+'!\"\"F2" }
{TEXT -1 8 "  and   " }{XPPEDIT 18 0 "x(0) = 50;" "6#/-%\"xG6#\"\"!\"#
]" }{TEXT -1 15 ". Solve to find" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
51 "dsolve(\{D(x)(t) = -2*x(t)/600 ,  x(0) = 50\}, x(t));" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,$-%$expG6#,$F'#!\"\"\"$+$\"#]" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "25 = rhs(%);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/\"#D,$-%$expG6#,$%\"tG#!\"\"\"$+$\"#]" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "solve(%, t);  #Find  t  in m
inutes" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%#lnG6#\"\"#\"$+$" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "%/60;   #Find  t  in hours" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%#lnG6#\"\"#\"\"&" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 472 38 "18. Solve \+
the initial value problem   " }{XPPEDIT 545 1 "dx/dt = t*sqrt(1-x^2),x
(0) = 0;" "6$/*&%#dxG\"\"\"%#dtG!\"\"*&%\"tGF&-%%sqrtG6#,&F&F&*$%\"xG
\"\"#F(F&/-F06#\"\"!F5" }{TEXT 544 12 ".   What is " }{XPPEDIT 551 1 "
x(10);" "6#-%\"xG6#\"#5" }{TEXT 550 68 "? \na) 0                 b) 1 \+
               c) sin(10)           d) " }{XPPEDIT 548 1 "sin(10)/2;" 
"6#*&-%$sinG6#\"#5\"\"\"\"\"#!\"\"" }{TEXT 546 27 "          e) sin(50
)   \nf) " }{XPPEDIT 549 1 "sin(50)/2;" "6#*&-%$sinG6#\"#]\"\"\"\"\"#!
\"\"" }{TEXT 547 78 "       g)  sin(100)     h) sin(100)/2      i) arc
sin(10)        j) arcsin(100)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 602 9 "Solution:
" }{TEXT 603 17 "   (b)    (0.68%)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 41 "Look at the differential equation. For   \+
" }{XPPEDIT 19 1 "0 < t;" "6#2\"\"!%\"tG" }{TEXT -1 15 "   we see that
 " }{XPPEDIT 19 1 "0 <= dx/dt;" "6#1\"\"!*&%#dxG\"\"\"%#dtG!\"\"" }
{TEXT -1 36 ".  It follows that x  is increasing." }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 57 "dsolve(\{diff(x(t),t) = t*sqrt(1-x(t)^2), x(0
) = 0\},x(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG-%$sin
G6#,$*$)F'\"\"#\"\"\"#F/F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 6 "When  " }{XPPEDIT 19 1 "t = sqrt(Pi);" "6#/%\"tG
-%%sqrtG6#%#PiG" }{TEXT -1 11 "  we have  " }{XPPEDIT 19 1 "x(t) = 1;
" "6#/-%\"xG6#%\"tG\"\"\"" }{TEXT -1 120 ".  This is the largest value
 for sine. Since x is increasing it follows that x must remain 1 once \+
it reaches that value." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 43 "Here is what the solution curve looks like:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "leftPart := plot(sin(1/2*t^2
), t = 0 .. sqrt(Pi)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "r
ightPart := plot(1, t = sqrt(Pi) .. 10):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "with(plots):\ndisplay(\{leftPart,rightPart\});\n" }}
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zFiz-%+AXESLABELSG6$Q\"t6\"Q!6\"-%%VIEWG6$;F($F^[lF)%(DEFAULTG" 1 2 0 
1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" 
}}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 473 172 "19.  Euler's Me
thod with step size 1 is used to approximate the solution of the initi
al value problem    dy/dx  =  y - x , y(2) = 5.   What is the approxim
ation for  y(4) ?" }}{PARA 0 "" 0 "" {TEXT 474 0 "" }}{PARA 0 "" 0 "" 
{TEXT 475 149 "a)  4          b)  5          c)  6            d)  7   \+
       e) 8               \nf)  9           g) 10         h)  11      \+
     i)  12         j) 13" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 604 9 "Solution:" }{TEXT 
605 14 "   (j)   (63%)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "x[0] = 2,y[0] = 5;
" "6$/&%\"xG6#\"\"!\"\"#/&%\"yG6#F'\"\"&" }}{PARA 0 "" 0 "" {XPPEDIT 
18 0 "x[1] = 3,y[1] = 5+1*(5-2);" "6$/&%\"xG6#\"\"\"\"\"$/&%\"yG6#F',&
\"\"&F'*&F'F',&F.F'\"\"#!\"\"F'F'" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "x[
2] = 4,y[2] = 8+1*(8-3);" "6$/&%\"xG6#\"\"#\"\"%/&%\"yG6#F',&\"\")\"\"
\"*&F/F/,&F.F/\"\"$!\"\"F/F/" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 484 180 "20.  Which of the follo
wing initial value problems is not guaranteed to\nhave a unique soluti
on by the existence-uniqueness theorem for first\norder differential e
quations.\n\n(I)     " }{XPPEDIT 19 1 "dy/dx = x^(1/3)+y,y(0) = 0;" "6
$/*&%#dyG\"\"\"%#dxG!\"\",&)%\"xG*&F&F&\"\"$F(F&%\"yGF&/-F.6#\"\"!F2" 
}{TEXT 504 9 "\n(II)    " }{XPPEDIT 19 1 "dy/dx = x+y^(1/3),y(0) = 0;
" "6$/*&%#dyG\"\"\"%#dxG!\"\",&%\"xGF&)%\"yG*&F&F&\"\"$F(F&/-F,6#\"\"!
F2" }{TEXT 505 14 "\n    \n(III)   " }{XPPEDIT 19 1 "dy/dx = ln(x^2+y^
2),y(0) = 0;" "6$/*&%#dyG\"\"\"%#dxG!\"\"-%#lnG6#,&*$%\"xG\"\"#F&*$%\"
yGF/F&/-F16#\"\"!F5" }{TEXT 506 1 "\n" }{TEXT -1 0 "" }}{PARA 3 "" 0 "
" {TEXT 485 232 "a) I only                   b)  II only \nc) III only
                d)  I and II only\ne)  I and III only      f)  II and \+
III only\ng)  I, II, and III        h)  All have unique solutions \ni)
  Wrong answer.     j)  Bonus wrong answer." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 596 9 
"Solution:" }{TEXT 597 15 "   (f)    (17%)" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let  " }{XPPEDIT 19 1 "f(x,y);" 
"6#-%\"fG6$%\"xG%\"yG" }{TEXT -1 71 "  denote the right side.  Our exi
stence-uniqueness theorem holds when  " }{XPPEDIT 19 1 "f(x,y);" "6#-%
\"fG6$%\"xG%\"yG" }{TEXT -1 6 "  and " }{XPPEDIT 19 1 "diff(f(x, y),y)
;" "6#-%%diffG6$-%\"fG6$%\"xG%\"yGF*" }{TEXT -1 116 " are continuous  \+
in a disk centered at the point  (0,0).   Equation I  meets these cond
itions but II and III do not." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}
{MARK "1 8 0" 321 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 
2 33 1 1 }