MATH
midtwo

# midtwo -

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Unformatted text preview: La [5 pts.] Solve the initial value problem, I w - X € 1 ﬁﬂai/ —; ‘C a6 \ MA .0 ﬂ C (Ye / w te M, m 5/” i ‘5, x12, ‘1 te - C at (L N X{0\:7% 7:«[+0 :{C'Lyﬁq 1“! XP/ %C{”r"+‘é 0% ‘ ,, ,3; m ‘1‘}: ﬁ~€:d’f'{:o—t+7§t Lb [5 pts.] Find the general solution of, (2t2x — 3)da: + (2m2 + 4)dt : 0. M a (My 1. M t X m 1? : tzx1«3x + m.) m . I \ LII/t —_ 215%? + lCﬂ : 215%”? L6 .1 ill I“) \$69} 1 £563} ; Ut 2. [10 pts.] A mass weighing 2 pounds stretches a spring .5 ft. At t = 0 the mass is released from a point 2/3 ft below the equilibrium position with an upward velocity of 4/3 ft / sec. Determine the equation of free motion. F; k“: ¥\lut‘\"\ FZMQ Z “95 "' kjm xx 2N0? '3» W\ 32 “PL/5:65" i / i : Lt Mai m3 N 3;“ s: K a ““““““ _ H m ' ¥~\\1/ ‘(9 AW“? ; FA‘j/l L1) I {i} X -1? r: i YA if j r / _ <.”~_____.M“MW__,,4___<~._,__~__WV._._.,.V_-_.—r’~~ , . \\ r skim 53.; ill 09x -/< ‘ ><l(0\‘ ' U/ ’ W 1, + f X : O J v 3 f 611 m 2 2 i ‘2’ K a ‘3 1 ”/3 (i. V t; O ‘ ”wan.“ "t M “.3." _ ﬁnﬂwﬂ ____é__h -_ u f 7,‘ cl ”f I ,/' M» J‘ W X 5/ ‘l ' ‘ //”’ / Cl I g or (L) x , , .~————~ M" / r 1 + e a; = O A, A_ it- ("27‘ , \F/ (QL/ (”L T— : 8 e (h m /{T‘: ClCo'i Z't 3" (1‘95 3'6. /’ a 2..) X(o>: C4 fa<(o\ 4 5’ 2 L/j ""9 C" h ’3 / \ v xﬁ-ti: *30i5im8'c 4' 362 Coe‘éﬁ‘f ‘I ( ”a /!\ i y (0} 1 {/5 t (f Cd (”:35 ’4: r ‘1', “z ”a 3.a [3 pts.] Determine a suitable form for the particual solution, yp(t), using the method of undetermined coefﬁcients, w+w+n=mmwm—w%¢ﬁ V (‘4 “:2 ”L. (A‘C‘t 8:} Si A L’f ’l’ ’YLt:+ El (“:33 git 3.b [7 pts.] Find the general solution to the differential equation, 9y” +6yl +y : 0. ”w (r x ‘ ("7 \r ’ M m r , w i J Ear 5e» \ 1, W a ' “1(4) , (L (‘\’L ; ,‘é. ; 3 chem¥\xpol “90“": (ll ,,_,» " likiN“? _\\$f; ' 3%; f/ 4. [10 pts.] Use reduction of order to ﬁnd a second solution of, \“ WM )2!” MW — 221 = 0, dw‘ge, if y1(a: )zfg'fw is one solution. 3X Y ‘ . c ( "\{"K.X} t. k “(7 ,5 ‘12:}: 2 um, mm \28M: 0 K“ ,MXV‘X 1 » I YIN): (My) Y1! "r (A \l (k): (M70 + (A (X) ‘l. '(Xkﬂ “mug y; (x33 (A (x? + u a"; ,frA 4» y'H’V) H ”()0 \{bq/OX’NLK 2/ , “2M( {Y} +[x+\\ “”(ﬂ , ”if 1‘ (\er'XZ/>(ZUKI(K>4 (y+\\3’q /r)+1(l+y\)</WY+UCJ()<>:'(% ' ~ “2(MW4'9 wﬂjacﬂ +( “72% X “Mm; (A (r) + zz’VWY/{AFX MOO 2- O M m 1+ m” Jaxw’ m =— 9 _ W MAW v3» MI WV +1 ((wérwwzjfwa} VIZ'CJ 645(19de ,/ “Liraw‘v- [Ww V‘Ef’w‘; 24,! z ’w dx i \ 2—. max“ \/ w 04(qu 2’ (3) ‘ I k .4, 1242" % /‘::, {4 -::;.f 7 ”X L I! 2 I ( 2 H .J V 1/4 Y m :0 {MW/[V‘X’C @(«Mx +9 / w (i C I 4"! V: Elf -—{7 (.4 / ,2: / / X X l 5 [10 pts.] Find the general solution of the differential equation, 2:" — 523' + 4x 2 864‘. ‘ PK.“ ;’ U't‘ f r 45 v’u \ f “L‘fw'a N‘W‘f‘aw 3 I.“ r v 90 C 55 "K ad” a a» A) “F Axlx W VM} ’3 ’ i We 4;“.ng M: em V}W€:ﬁ._ g M My w a M if; Exﬂ‘k ‘2‘ w“ b\ ”C “6’ \ , :L) 5* ‘- C3 J < T3 -. ’ d Cw 1‘7!” arm”; » s ,5” :3 1"“ w » H" 'z‘ 5“"; ~* uh: ’3 p , A M f m him” . 4. L4" _ ﬂ 4.»; 3 r» (it “C? 3 1:. R 2:731" w .5: g3 I» 5;}; l?’ a; X . a" .0; r ’ P 3 . L2) \n w k \ y(‘?ﬁ*’/h"/w WM ﬂ 1"“ , ,7 / ; 4 q f g! a; -‘ , f” / Xﬂg; 1* Cw F“ g ’2 ’" ‘ g "” i 6. [10 pts.] Use the Laplace transform to solve the IVP, y” — 43/ + 4y = 0, 11(0) = 1, y’(0) =ﬁ— W1 KW}:— ﬁ 315/} 4' W) ‘ ‘ ,4 p .5 Kt» , a H ; W - ‘v Q) ‘2 "'" ’3 Z ﬂ ’ '4' {4’ x ' at» x. .x' 6‘ / \ f f \l 4:» rl/ // v . ,~-~-_:-‘.___‘.,N; .V. , k.— ., ‘ ﬁ / ‘7 , \ , /’.‘ ., ( ‘bf r» .. ” 1"" 'n g ’8' f! 427‘ I” n: t?" J! ’ (() ) Lf « J m 5:,\,"« ”1 m i, _ , , 1 I 2.7191) 9 L (M .44 A\2?/ » ; "zI \1: L ENE); ¢ég 2 s «a; w d: \f‘ 7. [10 pts.] Using the deﬁnition of the Laplace transform, ﬁnd the Laplace transform of the given function, >4. f(t)=te3t a) +1 ‘qu finicfﬁt a x ’36 i?«{&>€; S t. {ijd‘& 5736 / O ,7 ”0 «37C at, I 2 4 m CH: ’ <30 r I t” C“ Q (7 Q) 02\$ ; (‘6 I, ", . ML: '6 U" éS'SJ—éfc .. Q C 5‘ 5/ i , , .. Jr ’C Q; J 1:“) dc.:; dc U ; g {v.59}; .2 . r 7 3~5 “ m 7 “’3 +7.5. 7 7% b-poo a t r“ 775\$ w (myb ‘5 < ("am \‘ 7 i we» Mb (1 ' W, C 3:"; 5; «ft: \_ 37+??? «4-? “)9 \ J «4 Q (l) \ x , Q ’3 3‘39 . o :7" a w 1'92! ---I ,. W ’ C 7279740 (3+5? {7 a»: x " \ (q .P g 36 f .4.” K} CK Z i; «r; a t, +7 7 ,7 3+» ...
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