v ~t't ..- l - uakron.edukreider/ode/sample3key.pdf · solve the initial value problem y"...
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I Test Tot," I Name _______________________________
Exam 3 Ordinary Differential Equations Dr. Kreider 23 Nov 2009 For full credit, show your work and use correct notation
1. Solve the initial value problem y' + 4y = 1, y(O) = 937 using Laplace Transforms.
..l...(sY-~~1-)+ 4-Y = S
V ~ ~q. ~ \ +~T't S (St-Lt)
~31- [ 1('1+ "..- l~+-Lt s S+-'t
- If!::.L::. ~3-=t- e- Itt + 1..'t l -t:; ) .eIf '-t
+
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2. Solve the initial value problem y" +4y' + 29y = 0, y(O) = 3, y'(O) = 4 using Laplace Transforms.
( "='.... y - ~S - 4-) + Lt ( s Y - 3. ') + 2 L? Y-=-O
3 ( S +- '2..) + ID
(~t-1...)2.. +- 2.5
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3. Solve the initial value problem y" + 9y = sin(3t), y(O) = 1,y'(O) = 0 using Laplace Transforms. 2ks
#22 tsin(kt) < - > (s2 + k 2)2
2k3
#25 sin(kt) - ktcos(kt) < - > (s2 + k2)2
(~'l.y_ s') -\- ~y -= -:; "1.
~ +- ~
y == S ~ ~Lf" "1. + -S+-~ (s'2.... t?J)"1. - Sy.
..., It ') -= c...os "1t + ..L LS~V\ ~t - '"3,t u>s"3:.t ]18'
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4. Solve the initial value problem y" + 3y' + 2y = 1 - U(t - 2), y(O) = 1,y'(0) = 0 using Laplace Transforms.
').. \ ) I -2~ cOPts(s 't"-S) ~:,(sy-\ -r'l'( ~ l - e ~ So
'2... ( S t" ~S -+ ~) '( = S+~ +...L. -L -'2..~
~ - ~ e (s+"2.. ') ('50+ I)
-2.-ss+>'( + e (~+2')(~+ I) Sbi-'2...)(~+I) S(~+'2...)(!.+I)
~~----------------~~ .u, ~,~ DV1.~ -h:> ~ e.-~'c.~t-
S+3 A-::. 13+ -::. ~ (~+, ') t- ~(~+'2- ') (~+"l.-} (5. + ~) t;+"1- <;. +- 1
S+~
::: D +-6-=- ~ - I "1.
'S-:.-'1- = -A +0\ ~
- ...L -\-2- - '2t -t> - e.. +- 1..e...'::.+'1-- -:='+1
LA +D+-"" PI -:. '/'l-.\ -::.
c... -:. - Ic. (-, ') [ , ')0+0-+I -=~'" 1["2
\ ..... ~ l-1.-) (-'J -t'D
~ 0
_ t _ '2.:t\ l'l.. '12 ..L .l- -L e e..r - + 'l-'> '2..<;:,+1.- S-+I
,
]_t- 'l.:t"L .L+ G- e"21
_1.-[-t:--'1-) - l + -'L) J )1- ~
.L G -L 'Ult--2.. 1..1..
[ ]'U[t- L )
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5. Solve the initial value problem y" - 6y' -7y = 4o(t - 3), y(O) = O,y'(O) = 8 using Laplace Transforms.
~ ( S'1- '( _ <3 ) - (., ( ~Y - D) - :t- y = 4- e - ~S ~
(s'l-_ ~'S. - 1- ') Y - <g + 4-.e -~~
Cs - =t-1CS+-'")
4--
'3 t- e... -~S
Y CC;-::r )lSt-I.J (<> - -:r- )S+ \ J ")
~ J.c ',-+ DVLLR,
-.L... A ~ (S - T )( S +- I "')
- i-60-1- 'S-\-I
l-=- f\ (t::.+- t ') +~C£.-T ')
s-- q - tt[;,A- +Dl
-+ e.(-~ )- l?c,. -::.. - I
l(~ G, -:::: (~)+
~+ I-;-1
'( -
't tt: '> -:..
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6. Solve the initial value problem using Laplace Transforms:
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x'(t) =
y'(t) = 3x(t)
4x(t)
4y(t)
7y(t)
x(O) = 3
y(O) = 9
For reference, the inverse of [ae b] is _1_ [d ad be
d -e
sX - ~ = ?'X-tt'(
s'(-':) = /.t){--=t-Y
'2... _ 5 f-L{-::,-S
(. ')(
-= L-:'f-S)[S-l)
~S - It; l [ ~ s - tS jJ
A)<. t- ~ <;,.+S <;,.-1
-l'l- ~ 0 +-~e,
- '3~ -- - '- Pr + D
s x=- ~+-s ~-\
_st t-=. S e- - '2-e
~~ - tS +y - L<;t-S ':L~-~)
_t..t> -::.. _~A-+O
_(,,:::' O-\-bQ,
..L ~ - \
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