3. Numerical Methods. Using Improved Euler method, solve for yı if x1 = 1.0:yy' = ² + e-x²2y(0) = 0.2n = 10Iteration Table:nXnk₁k₂Yn

Given Data: Let us consider the given data, y'=y2+e-x2y0=0.2n=0x1=1

Answered: 3. Numerical Methods. Using Improved…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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5) Solve the following problem, provide a well explained, correct and readable solution.
Apply the improved Euler method to approximate the solution on the interval [0, 0.5] with step size h= 0.1. Construct a table showing values of the approximate solution and the actual solution at the points x 0.1, 0.2, 0.3, 0.4, 0.5. y =y-x-3, y(0) = 1; y(x) = 4 +x-3 e* Complete the table below. (Round to four decimal places as needed.) 0.1 0.2 0.3 0.4 0.5 Actual, y (Xn) Improved Euler, yn
d) Applying four iterations of the Interval Halving Method to f(t) = p(t)-50, taking the initial approximations t₁ = 1.5 and t₂ = 3.5, yield the following approximations for the number of weeks after which Sibaliwe is able to recall exactly 50% of the proofs she memorized for the examination: = (2) 13 (4)
A hand-held calculator will suffice for the Problem. The problem an initial value problem and its exact solution are given. Approximate the values of x(0.2) and y(0.2)in three ways: (a) by the Euler method with two steps of size h = 0.1; (b) by the improved Euler method with a single step of size h = 0.2; and (c) by the Runge–Kutta method with a single step of size h = 0.2. Compare the approximate values with the actual values x(0.2) and y(0.2). x' = x -2y, x(0) = 0, y' = 2x + y, y(0) = 4; x(t) = -4et sin 2t, y(t) = 4et cos 2t
Q4. Solve numerically xy" + y + xy = 0 with y(1) = 0.7652, y'(1) = -0.44 and h = 0.25 (do 4 iterations)
In the initial value problem, use Euler's method with step sizes h = 0.1, 0.02, 0.004, and 0.0008 to approximate to four decimal places the values of the solution at ten equally spaced points of the given interval. y' = x + 4√√y, y(0) = 6,0 ≤× ≤2
2. An initial value problem and its exact solution y(x) are given below. Apply Euler's meu twice to approximate to this solution on the interval [0, ½], first with step size h = 0 25 (two steps), then with step size h = 0.1 (five steps). Compare the three-decimal-place values of the two approximations at x = 2 with the value y(½) of the actual solution. y' =(1+y?),y(0) = 1; y(x) : = tan (x + T) "ехact" approx. h f(xn, Yn) Yn+1 Xn+1 f(xn Yn) in Yn Yn+1 1 1 4

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3. Numerical Methods. Using Improved Euler method, solve for yı if x₁ =1.0: y = ² + e-x² y(0) = 0.2 n = 10 Iteration Table: n Xn k₁ k₂ Yn