Use Euler method to solve $y' = xy - y^2$, $y(0) = 2$ from 0 to 1, $h = 0.1$ Solve using Euler, Improved Euler and Runge-Kutta Methods. Comment. What is observed from graphs and data.
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We are asked to solve the equation from $x = 0$ to $x = 1$ with a step size of $h = 0.1$. Show more…
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i) Use the improved Euler's method to obtain the approximate value of y(1.2) for solution the differential equation y' = 2xy, for initial value y(1) = 1. Compare the results for h=0.1 and h=0.05. ii) Use the improved Euler's method to obtain the approximate value of y(0.3) for solution the differential equation y' = 1 + y^2, for initial value y(0) = 0. Compare the results for h=0.1 and h=0.05.
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Given an initial value problem y'(x) = x - y, y(0) = 1. (a) Find the approximate value of y(0.2) using (i) Euler's method with step size h = 0.1. (ii) Euler's method with step size h = 0.05. (iii) Improved Euler's method with step size h = 0.1. (b) Given that the exact solution is y(x) = 2e^{-x} + x - 1, find the absolute error for the results in (a). Compare and give comments about the order of the local truncation errors and number of evaluations per step for the methods that used.
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