Question

Required information Consider the following equation: dạy dt2 +9y=0 Given the initial conditions, 10) = 1 and y(0) = 0 and a step size = 0.1. Solve the given initial-value problem from t= 0 to 4 using Euler's method. (Round the final answers to four decimal places.) The solutions are as follows: t y z 0.1 1.2 2.3 4

Answer 1

First solve the given diffential equation and use given initial conditions to find constants involved consider y' i.e first derivation of the solution to be f(t,y) and then apply euler method ( kindly check attached images ) .

dy dt 2 +9y say, at -- 0 + 9)y=0 04+9=0 D-9 D = I 31 y (ta (CoA 3t+ Ce sin 3t using conditions, glo) = 1 G coso + Casino G=) and so, y 'Col=0 8'(+) = -Ų Sin 37.3 + 62 Con 37.3 34 sin 3t + 3 C2 copt V = -30samo + 3 C copo 3 6230 122=0

from 0 0 and 0 we have y (t) = con at (t) = - 3 sin 3t y and =f(ty). for enleis culer's method. dy f(ty) y (tol=yo dt then eulee method States that Yutt Inth. f(th, Yn) So, for and to =0 yo (tu=y(0:1) = 4y = Yo thif (fo-yo) = 1 + 0.1 F601) = l +0.1 6-3 sino? for He0.1 y 1 for t = 1.2 y (1.2) = Y = y + h f(ti,y) = 1 + 0.1 flot = 1 +0(-3 sinco1). = + (0.1(-3 %0.0017) 1-0.00051 0.9994 -

11 for t3 = 2.3 y (2. 3) = Yg = y2 + hf (+2, ₂) - 0.9994 + 0.1 f61.2,0.9994) = 0.9994 + 0.11 Sin (1.2) (3)) 0.9994 0.0062 [ya = 0.9931] for ty ty = 4 y (4) = Y y = Yth f (+3. Y3) = 0.9931 +0.1[2.3,0-9931) = 0.9931 to. [-38in 2-37 = 0.9931 + 0.1 [-330.040 -0.9931 -0.0120 [ Yu = 0.9811]