Math 222 EXAM I, September 26, 2007
Read each problem carefully. Show all your work for each problem. No Calculators!
1. (a) (8) Determine the order of each differential equation and whether it is
linear or nonlinear:
(i) (ii) > C >Ð" C ÑC œ #> 68 >ß .CÎ.> / Ð> "ÑÎC œ !
32ww w >
(b) (8) Solve the IVP : 6 , , and find theC C C œ ! CÐ!Ñ œ " C Ð!Ñ œ 5
ww w w
value of for which the solution approaches zero as 5>Ä_Þ
#Þ À >/ C œ / " ß > ! (a) (8) Solve the differential equation Cw C
(b) (8) Determine all values of for which the differential equation<
1Ð>"Ñ C #C œ!ß > ß
#ww
has a solution of the form .CœÐ>"Ñ
<
$Ñ + ,. (a (8) Find constants and such that all solutions of the differential equation
converge to as C œ +C , C œ "Î# > Ä _
w
(b (8) Write a differential equation whose general solution isÑ
Cœ-/ -/
"#
*> #>
4. (a ( ) Determine (without solving the problem) the longest intervalin whichÑ)
the following IVP is certain to have a unique solution, according to the
Existence and Uniqueness Theorem for linear equations. State reason for
your answer.
sec t 0 5Ð> *ÑC C 68Ð> "Ñ œ # ß CÐ Ñ œ
#w
(b) (8) Solve the differential equation À.CÎ.>> Cœ-9=Ð>Ñß>!
"
5. (a) (10) Solve the IVP: Cœ/ßCÐ!Ñœ!
w#C
(b) (8) Determine the interval in which the solution exists.
6. (a) (10) Solve the IVP: 2 4CC œ!ßCÐ!Ñœ!
w#
(b) (8) Determine the behavior of the solution for large t (i.e., find ).lim
>Ä_CÐ>Ñ