Math 222 EXAM II, October 24, 2007
Read each problem carefully. Show all your work for each problem. No Calculators!
1. (a (8) Determine if the functions are linearly dependent or independent:ÑCßC
"#
(i) t 1 2( t 1) , (ii) 3 + 3Cœllß Cœ Cœ>"ßCœ>
"# "#
(b (8) Find a function which satisfies the conditions: , Ñ 1ÐBÑ [ Ð0ß1Ñ œ B 0ÐBÑ œ BÞ
#Þ (a) (12) Use the method of undetermined coefficients to find a particular solution of the
differential equation
C C œ#/ ">
ww w >
(b) (6) Determine the general solution of the above equation
$Ñ Cœ/. (a (12) Given that is a solution of the differential equation
"B
,BC ÐB "ÑC C œ !ß B !
ww w
use the method of reduction of order to find the second linerlyindependent
solution y#Þ
(b (6) Determine the homogeneous ODE whose general solution isÑ
C œ - / - >/ / Ð- -9=#> - =38#>Ñ
"# $ %
>>>
4. ( 6) Use the method of variation of parameter to find a particular solution of"
the differential equation
#C %C #C œ / ß > !
ww w >
"
>
5. (16) Determine the form of particular solution of the following ODE, using
the method of undetermined coefficients. Do NOT evaluate the constants.
C #C #C œ %/ #/ -9=Ð>Ñ >/
Ð%Ñ Ð$Ñ ww > > >
6. (16) Solve the initial value problem
0C C C C œ !ß CÐ!Ñ œ #ß C Ð!Ñ œ "ß C Ð!Ñ œ
Ð$Ñ ww w w ww
