# MAD4401ReviewTest2 - dx dt = x t Ā sin t x(0 = 1 Problem...

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MAD 4401 Review for Test 2 James Keesling March 4, 2009 For each of the problems below, be sure that you get the correct answers and can explain the theory behind the calculations. 1 Theory of Diﬀerential Equations Problem 1.1. Sketch a proof that dx dt = f ( t,x ) x ( t 0 ) = x 0 has a solution using Picard Iteration. Problem 1.2. Do several iterations of Picard iteration for the following diﬀerential equa- tion. dx dt = t · x x (0) = 1 Problem 1.3. Do several iterations of Picard iteration for the following diﬀerential equa- tion. dx dt = t 2 · x x (0) = 1 Problem 1.4. Do several iterations of Picard iteration for the following diﬀerential equa- tion. dx dt = t · x 2 x (0) = 2 1

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2 Taylor Method Problem 2.1. Determine a power series solution to the following diﬀerential equation.
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Unformatted text preview: dx dt = x + t Ā· sin( t ) x (0) = 1 Problem 2.2. Determine a power series solution to the following diļ¬erential equation. dx dt = x x (0) = 1 Problem 2.3. Determine a power series solution to the following diļ¬erential equation. dx dt = x + t 2 Ā· x x (0) = 1 3 Numerical Methods: Taylor, Euler, Modiļ¬ed Euler, Heun, Runge-Kutta Problem 3.1. Solve the following diļ¬erential equations using the Taylor Method, the Euler Method, the Heun Method, and the Runge-Kutta Method. dx dt = t Ā· x x (0) = 1 dx dt = t Ā· sin( x ) x (0) = 1 dx dt = t 2 Ā· x x (0) = 1 dx dt = t Ā· x 2 x (0) = 1 2...
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MAD4401ReviewTest2 - dx dt = x t Ā sin t x(0 = 1 Problem...

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