In this question we will show that Euler's method is a first order scheme. Consider the ODE y' = f(t,y) , y(t0) = y0 where we assume that f, ...

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3. (25 points) In this question we will show that Euler's method is a rst order scheme. Consider the ODE {y; = f\": y): 1930) = 90: where we assume that f, g, 3: are continuous. Let h > 0 and yn+1 = y\" + fan: 3/\") ' h: be the Euler approximation at time tn+1 = in + h. Recall that the global truncation error is dened to be En = Eln) _ yu- (a) Using Taylor's Theorem, show that 1 A En+1 = En 'l' (9'8\") _ fan: 911)) 'h 'l' 59'1\"\") ' '12: 5f g L. Using Taylor's Theorem applied (b) Suppose that for all (t, y) we have the estimate 8y to the function g(y) = f(tn,y) show that lyqtn) _ fan: WI 3 LlEnl- (0) Suppose that for all t we also have the estimate |y"| g ZB. Using parts (a) and (b) show that |aa3(r+mmaq+3m. ((1) Suppose that we have the estimate u+hmn1 () IEnI s L Bh. Using part (c) show that we must then have (1 + hL)"+1 1 L (e) As E0 = 0 satises (), we can then use part ((1) to conclude that (69) is true for all n 2 0. (This is called \"mathematical induction\".) Using that E1n satises the estimate () for all n. 2 0 and recalling the fact that IEn+1| S Bh. show that