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1. Write a description the Euler method and the implicit trapezoid scheme, what they is doing, and why. In particular, give the derivations of these methods. Describe the idea used to get Runge-Kutta type algorithins. Write at a level in this question and, indeed, for all questions in this assignment and your other assignments in this class, so that a junior in college with the prerequisites for this class can understand what you are saying. OF course, for each of the problems below, please tell me what you think is happening. 2.Consider the initial value problem dy = dt y (0) = 0 For the interval [0, 1], determine how small of an h needs to be used to in the Euler method to obtain an error of less than 10-3. Using this h, approximate the solution to this problem using Euler's method and plot the result against the true solution. Do the same with the Matlab command ode45. 3. Consider the initial value problem x 1 = -64 ; x y : 1 (0) - Using this h = .1 approximate the solution to this problem using Euler's method and plot the result for IF against the true solution, cos (8t). Do the same with the Matlab command odc45, setting tspan-[01]/ 4. Consider the initial value problem -50 (1 + 10t) 0.1 x - y 1 -t y (0) = : 1 Using this h = .1 approximate the solution to this problem using Euler's method and plot the result for x. Do the same with the Matlab commands ode45 and odel5s, setting tspan-[01]. 1

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Differential Equation Solution Methods
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