Midterm1.pdf

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• Test Prep
• 7

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1. (10 pts) (a) Find the explicit solution b #: cos(t)(E + 3r2) with g(0) : 1- tepse*ion { s-.riAJes .^e-{hJ' l+:- =f.*1 J+ =) J 0 t)&J=l^n Jjct *)) -* F,.lo\ R..s,-r\.. \ , A-1 R Ntr*.J- B3:-\ .1 [\".{ ) t 1 1::,-r=., no*=-'^ , a*.' 'lr =1 Ln\3\ -An\ l*y\ = s*{ + >, [. Lfu\ =-\n\+e st {< Tl..,d'' lfar e* =t-+- 2 Y.* I )rr\ =\ .{", \n l. : g '= n" K e\*{- - =r 5 -r t: .d"! *cr f =-) C =J-. "\ = Find the general solution to * : "" - 10. drr condition so that the solution is unique. sx Ar .h r- \ .q : €" ' tJ \l^.o O tr { (b) ?C r '\ €--k I t) I = x ^k ) 42- .)'"n " JK T.B.P, : Ken* fe-KJx , e initial (€ K * efu C Ytx\ A Ax Y
2. (5pts) (:^r)(;"r) Find ancl classify all equilibrium solutio n, rc f,: (10 -'a)'@' - 4). Elut\ibn\rrrt*tu-\,V'.s I :=\Q l:7 ) :-2 t \S t. 'z_* -2- ri N 'l J 4 I I I d ,\ \ i te*ir{*LL undoLL s\..Lt*

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3. (15 pts) (u) A baseball is dropped from an airplane. The mass of the baseball is about 0.2 kg. The force due to air resistance is proportional, and in opposite direction, to the velocity with proportionality constant k (where k > 0). Just like what we did in homework assume that there are two fbrces acting on the ball: the force due to gravity and the force due to air resistance. (Recall: Newton's second law is wLa: F and the acceleration due to gravity is 9.8 meters/second2). i. Give the differential dquation and the initial condition for the velocity u(t). (Do not solve) t ) r- Av = -tr.'-1 -K/ .,Etcl z-(i' =) t( O.
• Fall '08
• NAGY,KRISZTINA
• Constant of integration, Boundary value problem, Picard–Lindelöf theorem

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