MAP 2302 Quiz #3
Name:
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ID#
FrJ1 dOn
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~
_ __'__'___ _ _ _ __
HONOR CODE: On my honor, I have neither given nor received any aid on this
examination.
Signature: _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __
Instructions: Do all scratch work on the quiz itself. Make sure your final answers
are clearly labeled. Be sure to simplify all answers whenever possible. SHOW ALL
WORK ON THIS QUIZ IN ORDER TO RECEIVE FULL CREDIT!!!
No.
Score
1
/8
2
3
/14
I Total I
/8
/30
I
(1) Rewrite the given expression as a single power series whose general term involves
xk. (8 points)
=
L n(n -
l)cn x n + 2
=
L n(n -
1)en xn - 2 + 3
=
L nenxn
~~~
s+ruh
S-hvts ..,...w...): 1
00
~ t'\ (" " ) c..,X"+ 'tC
L \-
12(3 X~ ~
t.
u ·,.}t..
);0
'&f.tr-t-s wr+L.
~-'lc.'1X~' 2 -l- ~ C, ~ +:, ~,nG,x'"
t'\ C
~
00 n
~ S(t'\f2)(tI-l-i)C".. ;X
~c, \- (12(3+3(,)1< 4- t["C,,',)C,," U,,~;).l("H)CH +3V1 c...1x'"
t\-=l.
oX \
(2) Find a power series solution of the given differential equation. Identify the series
solution in terms of familiar elementary functions. (14 points)
y' = 4y
I' :: ~'1
~
k.
k.
1)0
Lk~x ·' ~ ~L C\t.X ~~,
~ ~~
~
00
t/c.~ i)(~~)(It", '(~
Cltfl
:::
L ::.
I
C1. ::
'-\(.;1.
3
\{(~
_ ~ . 4' \I,~
-
3·1.
_ 4. · \{ · \.( · ~(o
\:t ­
--
":} . (o
~, ~
\l
Il-
~
.
'T
1
I-x?
=- X-
1. '\
,'I... ~.
}
5~
s-:!. l
1-.
X - - , 'X - .' .
(3) Find two power series solutions of the given differential equation. (8 points)
y" - xy'
+ 2y =
0
00
Hint: After plugging in y =
L
CkXk
and its derivatives, the equation becomes
k=O
00
L
2C2 + 2ea +
[(k + 2)(k + 1)ck+2 - (k - 2)CkJ Xk
k=l
(kxi{~ti')CUt - (l~Z)C~ ::0
=- (~-Z,)CIc.
C
~'l
(It'ri)(k.~~
c~ =
C \I
=0
('5 -=(10 :;
c~
l) ~ \{
- --
~(::.1
- c.. \
'5 .'1 .\-1.
--
-::0
to ' )
c.,,"
--
3C.~
1 ·<0
~C.I
:j:f
..h
5~
=
0