MAP 2302 Exam #2
Name:
_-----,-_K--'
~wtr ~f---_ _ __
ID#
FJJ
loU
HONOR CODE: On my honor , I have neither given nor received any aid on this
examination.
Signature: _ _ _ _ __ _ _ _ _ _ _ _ _ _ __ _
Instructions: Do all scratch work on the test itself. Ma.ke sme your fin a.l answers
are clearly labelled . Be sure to simplify all answers whenever possible. SHOW ALL
WORK ON THIS EXAM IN ORDER TO RECEIVE FULL CREDIT"I
No.
Score
/ 16
/30
/45
1
2
3
I
4
Total
/9
I
/100
I
(1) Determine whether the given set of functions is linearly independent on the in­
terval (-00,00). (8 points each)
(a) fl(X) = X, h(x) = x 2, h(x) = 4x - 3x 2
W()/'(2./
~'(_3'(t)=
X
X-
J.
'(\(-lx':l
-::. ~
l{-Co)t
2~
0
(b) JI(x)
=
eX,
VI(t'(, to~ X ~\~ )c ) -­
e.)t
COSX,
~y.
=(,
X
l-Si l\
-(g
4-~x
0
- (,
h(x) = sinx
Cvr>'IC
f
eX
'l
~-!ok \ _ ....'
-~
7..
h(.1:) =
Z"
-SfV\ "Ie
-MX
~I"
:x
Qb)c
~,n)c
'Ie
- to'))c
-=-e\('l\'t\t~ ... ~l~)
~
===-\
--LOS 'X fe~Sf'f\)t-e.XC~sX) f-c;,,,
>t( -(.J(c.~s); +t~'5i" >t)
~ (~~-ht)
2)(
0 Z-­
(2) Find the general solution of each differential equation. (6 points each )
(a)
y(4) -
8y(3)
+ l 6y"
= 0
fY\'4- ~~~ -H~r/::-0
~t (-,,'L_~W\ f'fl,) =0
rn1. (rY\ - '1) (~-\{') :;'0
(b)
y (4)
P' '-i
+ l 8y" + 8ly = 0
+,
~ ~z
+~ (:::- V
(~~+~) (~2.tG)
iV\'l.-
~ =-0
'=0
\.,
~ -M
1.-
=-0
""
t-3'CI
,0, -- ­
(c) y"-6y'+13y=O
~y?- - Cot'h. +r3 -:::
t1\ -=- -
Q
(~) ± ~ {-~y - y (,)(~5 . -=
~(I)
- - _ ... ­
-
+ 3y' -
lOy = 0
V3b - S'l. ~
~(3:t: 2i-)
(d) y"
~:t
~y
(e) y"
+ 6y' + j! =
~2. +<OMt
0
'1=-0
(~~ 3)(~t'3\:;..a
i
(3) Find the general solution of each differential equation. (15 points each)
(a) y" - y' - 6y = 2sin3x
(~-3) (fY\~;}.) =-0
vY\-=-
~I
-;).
I
Y ':: 3Ac.os 3)(. - 3BcslW\3x
r
Yrl::
- 'ASi"~)(
- qBtbS)x
yi'"'~f' - ~7r :' - 1A~\n 3" - , &eos ~x -=
-q~t~~k - 't&(.05~X
(~ACA., h:- ~~ s\~ ~)c) - ~(As\i\ ~ ~ tBCo~ '3 ~)
- 3A-cos3)c; +-3B~""~x-CoA-sl~~)(-G,6{,os3x
~~
- 15t4 -\-3~~.l ~
~s-C- ~ A - ·15"'6 =-0 J
~
-,sf
t3B ~ ;1
I-slA t"1ffi -:=o
":}-C(&~
---18"1\ ~ I 0
A-- -lQ..
-:u -- - s:
3~
~
2
~_--L
-=- 'It - 3'\
(b) y" + 9y = 2 sec 3x
Lt.' -
~ 3-x . ~ _
3
1-
2.
3
~,-;: ~ JolJx:: ~ ~
Y= L. CoS 3)( +-C~~jV\~)c +- ~ I", \eos ~)C\. Cos h'
z.
+ "'\.\~I'" ~)c
lA -=
c.()~ ~)<.
c{,\A ':- - '3 S\~~xci X
- ~ct\.\:: <:'i~3'1(M
TJ-fA tk ~ ~ 1__1~ \
=:
~ I" \M ~ Ie \
(c) y"
1" ~ '0' -)
+ 2y' -
3y = 1 + xeX
:::- 0
-t-2(r-t- ~x~\l -)(A+(ti-~)/) fit"L -\- 2m - 3 :::- 0
(VV\~ 3) (fr\ -I) ~ () ~ ~ -x-e'" + (~St~t)e~ -~~ =- 1~ xe J(
-
~f? = t ~B~
M:: -"), \ -~
(C :: ~ e
~If• +'b/If'- 3IP=(D.../
+(~&tI\)c ... ~!,~~,9) e)(
\~ . ~ .~
'f..
-tC'le
~S : '(r -:: A-\- (eNc.~~ J-v:A 6vt5~', 1~
'" IH '( (6~~C.)e~ -= A+ (e,)t+ ex.)t
)pl :: (2-~x ~c)eX + (~\/"" ()c)e ~
-: . ( th C2BK\A~C)/
~; = (1. i1>H (z6t<-))l +("b(2&\{))<~/
:(6)('- -\- (I( e, -I-Lh f (211)-1- 2(~el\
~ et-\{ c.. -= 0
-3A
=:
I~
~f" - ~
-cl)
t
~ +-4. c.. =0 ~C -= -1[P
A-=-1
t{ 1't'-ii)r\lC
(4) Set up the appropriate form of a particular solution YP' but do not determine the
values of the coefficients. (9 points)
y(5) _
~ (S)
I
{3)
= eX
+ 2X2
(l)
-y
=-0
'(r:~ (~2. --,) =-0
'IY\?> (~t ,) (h\ - \) -=- 0
~..4# ~vf.~~;J ~r ~ 1+'1. ~ X -I-
'(."3
(C,/ ~D t.t- ~)
-, -
5
Variation
Scratch Paper