In class exam 3

# In class exam 3 - Yo 1;£ ~ T;f< ~ 7 ~ 37 =-~ e t-t 1 l...

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Fall 2008 Math 285 Exam 3 Version A: November 21 Name: f\",5 vU eM If you cannot complete a problem (perhaps because you forgot a formula) but you think you know how, please describe. Correct methods will receive partial credits. Be sure you have a copy of the Laplace Transform table. 1. Write the linear system of ordinary differential equations in matrix form. Please do not solve. ~~= x + 2y - z - 2e-t sin(2t) dy dt = 4x + 3z + e-t cos(2t) dz dt = y + 6z + e-t / -t. r-x .- '2. e.. ~ l~ L L ~ i- e---tWSLL \ b.t:- --r t: e . lOr-- X( ~.(~ ~ -~ ) X T ~ ~L-~'~~ \ IA/'l~ X ~ (~ \ o l b e-1:. ) t ) 2. Verify that the set of vectors Xl (t) = ~) e-3t and X2(t) = (~) te-3t + (~) e-3t . forms a linearly independent set on the interval (- 00, 00). 1 • I

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(-12t - i) 3. Verify that Xp = _~ 3 is a solution of the system of ODEs 2 • I i ~
5. Find the following Laplace Transforms using the table. (a) L:{e-2t-S} ~ 'e.-s ;t~ e -"l~ 1-:: e -s~ ."3t -1't. e, - e.. =- 2s,V\. ~ (~L) ,/ b £'--Cj. L~_4.~-+~ ~3 SL S '3 ~<.- 9. II ~ ~ '-.S,"" ~CSl-)\ ~ 2' (c) L:{ (2t - 1) 2} =- L~ 4:-"t ?..- 't- t>t" \ 1::: 4-. L-t"' sz. S 6. Find the following Inverse Laplace Transforms using the table. (a)L:-l{S~3+S::~S}::' L-l\-L+ gl 4 \ ( ( ~~ ~ %C.S~3) S(S"t 3) ( " --L-\

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Unformatted text preview: ? Yo': 1) + ;£ -( ~ +\ T ;f '< ? - ~ 7 . ~-+37 =-~ e..-] t-t 1-- l e..-1""( _ S--1\. l 3 3-3~ ""t 3 (b) C'{ :2S:1~} ,-'{L-, ) .~ ? ••. ';[ -, (l Y-~ S '"t l b 1 r e,. . s z,. l(. ., (-:: ~ c. .oS4 t-t-t ~>LlA 't L. 3 / • I 7. Using the Laplace transform, solve the following initial value problem in the Laplace space; that is, you do not have to take the inverse Laplace transform. y"-y = sin2t, y(O) = 1, y'(O) =-1 J\~e~ L ~ ~D,-L, s~~'>: ~ 1-i.~ j 1-s ~ LO)-Y( (0) -L) j~~ ~~ I-I ~~'--l)~ 1 J '1 ~ s - \-t _L_ \$L.~'t. ~o L1 j) ~ ~-l-+ ~ ~L-l ~ L, 't-X5 L_ l) ~ l-+ ~ S-t\ (<; L-t <to-x.£-l X-S"T l). Extra credit (up to 3 points): Solve the above IVP completely by taking the inverse Laplace transform. Note the partial fraction decomposition format p A8+B C---+--(82 + q) (8 + r) 82 + q 8 + r j l L') = e,-t + t ~ ,lA ( L L)-+ ~ e-t --t e -t ~ l-c) ~ ~-t--l t. \ ~ \:. .--t S e--t-S S,l/\. CL • I...
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In class exam 3 - Yo 1;£ ~ T;f< ~ 7 ~ 37 =-~ e t-t 1 l...

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