MAT$262$Differential$Equations$
Exam$1$$February$2024$$$$$$$$$$$$Name______________________________$
$
The$following$problems$are$a$group$take-home$portion$of$the$exam.$$This$portion$is$open-book,$
open$note.$$You$may$use$any$resource$to$solve$these$problems$(don’t$use$ChatGPT$or$some$other$
software$which$can$solve$the$problem$for$you).$You$may$work$together$in$groups$of$two$or$three.$$
Your$group$needs$to$turn$in$only$one$answer$sheet.$These$problems$are$due$on$Wednesday,$
February$28,$2024.$$Draw$boxes$around$your$answers.$$You$can$email$these$to$me$at$
pbialek@oakton.edu$$
$
1. (10)$Suppose$a$cup$of$coffee$is$185°$at$2:00.$At$2:09,$the$coffee$is$170°$F.$$If$the$coffee$cools$
according$to$Newton’s$Law$of$Cooling$and$the$air$temperature$is$68°$F,$determine$when$it$
will$be$our$desired$temperature$of$135°.$
$
2. (10)$Use$the$RK4$method$(fourth-order$Runge-Kutta$method)$to$obtain$a$four-decimal$
approximation$of$𝑦(0.1)-with$step$size$ℎ = 0.1$for$the$differential$equation.$$
$𝑦!= 6𝑥 − 8𝑦, 𝑦(0)= 2.$$You$can$check$your$answer$with$online$Runge-Kutta$method$
software,$but$you$also$need$to$do$the$problem$without$such$software$and$show$your$work.$$
$
3. (12)$Suppose$a$raindrop$with$a$mass$of$11$mg-falls$from$a$cloud$that$is$4500$m$high.$$Let
𝑥(𝑡)$represent$the$distance$the$object$has$fallen$in-𝑡$seconds.$Assume$that$the$force$of$air$
resistance$is$directly$proportional$to$the$velocity$𝑣(𝑡)$and$that$the$proportionality$
constant$𝑏 = 5.2 × 10"#-N-sec/m.$$
a. Find$an$equation$for$the$velocity$𝑣(𝑡).$
b. Find$an$equation$for$the$motion$𝑥(𝑡).$
c. Determine$when$the$raindrop$will$strike$the$ground.$$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
