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Math 106A&B INITIALS Exam I, page 1. aS September 29, 2006 — sam Giten ang pone (eos Ya) AS t { ae “nthe Edler melbol gppanmotion 4\\ — Hes Bone a ate tape bese He oot len fo) Got Heap fon Teliesyoat an enti J the dope tel a ie dlamutt: ot (tny Yu) ante ont ay tho egprtinee (taety You) 3 [hs Lele ' no NOE ! ff 7 required in or bbe] Spe ee che tee 2 1. The slope field for the differential equation dy/dt = (y+ 1)(¢ — sin 2t) is drawn above. Using only a gut ruler and pencil (no calculator values), sketch the Euler method solution through the point (—1,1) as far as possible, using stepsize h = 0.5; your sketch will be a collection of big dots connected by line segments. (see sketch L 2) dy/dt = (y + 1)(¢ — sin 2t) : . Consider the initial value problem { y(0) =2 (The solution is not the same as the one in problem 1 as it “starts” at a different point) 2a. Find the find the value of C such that y(t) = Ce®5(?+0082t) _ 1 is the solution of this IVP. Show all your work. (You don’t have to verify y(t) “satisfies” the DE; just find C) we. pec err, Bu == wt 1846... a Gece | = a-OE 2b. Use your calculator to find the first four Euler approximations to the solution of the [VP in (2), using a step size of A = 0.25. Organize your results in a table, writing all results to four decimal places. Compare the exact value of y(1) to its approximation in the table. How far apart are they? ‘\ ‘ 0.25 4H) ye (r41(T- sin(2r))H —> i repeat. the jo (este ca ae puter es 2 H— T | prodice (ea) : os(i+a52))_ .\PA% (Sy. - TABLE . ; Expr vive j, |, 8196 (e ) tte ahi Je O [2 ( stg point) = ee in He ayers a toe pe 1364-14265 c<- 0.0099 Loo | 4265 pettor. 4 YC) 
