MAP 2302 Exam #4

Name: (:m'tNJtf K~

ID# Fdl ~ol\

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. Make sure your final answers are clearly labelled. Be sure to simplify all answers whenever possible. SHOW ALL WORK ON THIS EXAM IN ORDER TO RECEIVE FULL CREDIT!!!

No. Score 1 /12 2 /24 3 /32 4 /32

I Total I /100 I

(1) Rewrite the given expression as a single power series whose general term involves xk. (12 points)

00 00 00

n+1L n(n - 1)Cnxn-

2 + L nCnxn+1 + L cnx

;::=2 __ -" ~~ _:·I~ '1.0 S~wi;t...X1

~~2 f (~C3tCo)X ~ t [(~H)(lf0C"'1- +- (I:.-~c.._, ~ c.-1Xk. ~-::l

(2) Suppose y = L00

Ckxk is the power series solution to the differential equation k=O

y" - 2xy' + y = o.

Find the recurrence relations which would allow you to determine all the Ck'S in terms of Co and Cl. DO NOT try to find the formula for Ck or solve the equation. (24 points)

0. k. t 00 .... - l 00 ~ ~2

~ -=. 5~CtXJ Y:;. E,~(lX YII ::: ~lk(~-')(~XI

f k.(1 -I)C~Xl.- 2 - 2x {: ~(lX~-t +- tc~Xl :=. O bt ~ ~

()o () ~ -l !>o l Po lL l l -\ CIL X - 2 L.. k;' X t L e X =- D ~ ,~ :;'l .r-/ ~ ~h ...,.it<.- ,<0 s\tvh c.;tL~' £, tr.r~ -.A'ti- ):0

(:U l +4)+ ~ [(l+l)(h~(~n - J,\t.Ck. 4-(~1 X~ =- 0

lUa.c.)~ ~t(k.~Mw)c..., -I- (1-2l)qX' =0

c :: - ~ 1. J.

Cn-t)4C - ­

~\l. '" (~+'D (~~1)

(3) Use the power series method to solve the given initial-value problem. Identify the series solution in terms of familiar elementary functions. (32 points)

y" + y' - 2y = 0; y(O) = I, y'(O) = -2

y' (Q) =C, =- J.

00 ~-2 'fll:: ~2.~(~ -()CIt ~

~ [ (hJ.)(~H)(~l -I- (~}f)(~.. - lCIo-1x~ ~O

(kt~)( ~f\) Lt-U .\- (~ ·H) c.~4'\ - J.L~ ::: 0

C := !L~ -(k.~I)CVI ~1. (k-+l) (k-+-\)

- - ---

(3) (continued)

~CO - I-e ) -1- 1-(-2)C -::L ­

~ . \ J . \

ac.,- 2Cz. :J.(-:l) - J(~)C - ­~-3-2. 3 -.;2.

.2+l- - .:1..- c;t • ( ­~

- \f- ~ <6 -3-z. 3-1.

-- - Ire _ - {b-2 - is! -3

3

5 - l( · ~ 5 -4.3'1

- e - l.)c

(4) Find two power series solution of the given differential equation. Write out the first four terms of each series solution. (32 points)

y" - (x + 1)y' - y = 0

~ C "l , Ot> . ~ - , I, ()c ) ~-l,=- ~ t." I y =L k. Cl X ) Y =L. k.(l -I t;. X ~ - v t.=-, ..... :: z..

~+C.lC'2 :: --­~( -L _f ~ O Z.

~ , "-0

Ctt-+(M-IC~1. :.

~ -\-7-

(4) (continued)

('1+(1 C,+ CoTC, ;)C., +Co +(,(3 :: ... - -2 C. -t3~

:: ~ ...­3 "3 =­~ Ol · ~

c.+<:-, C.f 3(,Cz +C) ~"t 3~ C.+1LIC'l=: = 2- ';2 . 3 := ~ .3- .}- ;t.J ~Cof~ ,

4 -- ~

I...{ \.t -- 4 '3 ·2 ­

"3..1. 3 \ Y \ \f" ~ C +-c, y + 1.C \,:Z + ..1. C \r"l -1 C ~ + ex+- - c X + 'iT C X +. ­I . , 0 1\, 01\ 1 ,I'- T (!) V Z , (g 0 "t I