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Exercise 4) The text homework problems 2.4.4, 2.5.4, 2.6.4 are canceled. However, you will work on a similar set of “by hand” problems in the lab. In preparation for that, use the Euler, Improved Euler, and Runge Kutta pseudo-code below to estimate e’, as the solution to the initial value problem that we’ve used exclusively in these notes, namely y’(x) y (In the • will be studying different initial value problems In this exercise use just one step, ii I, of si iso, use the slope fields to illustrate how you fin’ and use the various k values which are rates o c anges that get represented as slopes in the solution :raphs. Take one step to get from the initial point (x0,y0) (0, 1) to the final point (x1,y1 (I,?? ..: — ) V 4~ Euler pseudocode for y’ (x) f(x, y) ~Vo) Yo k f(x~i) x j+I x+h j y+hk. (answer is (x ~‘~) (i~ 2).) / V II 340-0 I I 3.5 II) 1? I k= çco, lii.’lTF Ill/li Ill I1I~iffIlttl I IlIl,,,,,, 111171 III? 3 y1’: ‘y04-kk 2.5 , , , I IlFIlFrl?1Il7P7l Ill / 1171171FF It 0~a fill /.‘7t1f117I17r1lI~ 2. ~2 t I /11/11/11/1111~ (1,1) E’$er. 1.5 - - - U. —05 1,0177, 1.7802 4b) Improved Euler pseudocode: (a, i’) k~ f(x3~) /c2 f(x+h,y+hk) k1t fit8 ‘)“l. k lC2t ~~(&,ItC)t x J ~-(k + 1c2) I x+h j y+hk (answer isy1 kt 4-(i~z) za k(I~l):I S 2.5) y1ty0+-k14 i. i (x~,y1) r c~,z.c) IS 4c). Runge-Kutta pseudocode: k1 (.5.Y) ~r h h) 2’ ‘ h h k3 4x+ 2’~ : Ic) Ic4 f(x+h~.v+hk\ t Ix k 2 ~J Ic x jI y J 1 j c(O~S y+hk Ij (‘~,~D’— 2.708) V =V (°,i.~ 4 kA.61~frnDt I I I a’ I /1~1111? I/stiff fit / 1 it/f If fitif/tIt k~t j-(,~+h,~-bk,) ~4 (.5, i~.c) 3.5 -1!? a’ I , I F I- t F S I I a’ I 7 1 1 5 I,, / I iItI,,ItI)’I,I,, . &. (oi.c, L~- .so.n) t LW~J~bIh4~ x+h j (answer by k,t cLo.~u. 3) 1 j I€#s.is4pofn1r ‘‘I /5 /,t,,,i,ItI,II? 3 £(.s,i.7s)tI.75 7~) 7, ii??? 7.7? a’, a’,?? a’ I 2.5 7. .7 1 1! iC4~• &(1, \4- L’~b1S) I,, ft / b ‘It it 1t V2 ‘it 7, 1t/F kt .7.’, j.7o7 .7, , 1.5 .7 .7. , ,_? ~ Ns..i,. .7 ‘ / , ~ A 1 ti 4-170%’ ——A 2-’ot.. AA’AAAAAAA,’ 0.5 (x11l~1”yt (t,t.7ot. 0 e. et.f~A,J[~ ~t4s 2 ]It.. -0.5 1.0177, 1.7802 0 1.5 x 1~