HW_07_Solutions - A Problem 1 Eulers method(a The true...

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A. Problem 1. Euler’s method (a) The true solution, Y(x), is only given as a reference for the obtained numerical solutions. Thus, we have to avoid any kind of substitutions of the given true soution into Euler equation or the given differential equation. Y 0 = [ cos ( Y ( x ))] 2 , 0 x 0 . 5 , Y (0) = 0 , h = 0 . 1 Y ( x ) = tan - 1 ( x ) x n +1 = x n + h, y n +1 = y n + hy 0 ( x n , y n ) = y n + h [ cos ( y ( x n ))] 2 , Y ( x ) = tan - 1 ( x ) x 0 = 0 , y 0 = 0 , error = y ( x n ) - Y ( x n ) x 1 = 0 . 1 , y 1 = 0 + 0 . 1 · cos 2 (0) = 0 . 1 , Y (0 . 1) = 0 . 0997 , error = 0 . 0003 x 2 = 0 . 2 , y 2 = 0 . 1 + 0 . 1 · cos 2 (0 . 1) = 0 . 199003 , Y (0 . 2) = 0 . 197 , error = 0 . 002003 x 3 = 0 . 3 , y 3 = 0 . 199003 + 0 . 1 · cos 2 (0 . 199003) = 0 . 295095 , Y (0 . 3) = 0 . 291 , error = 0 . 004095 x 4 = 0 . 4 , y 4 = 0 . 295095 + 0 . 1 · cos 2 (0 . 295095) = 0 . 386637 , Y (0 . 4) = 0 . 381 , error = 0 . 005637 x 5 = 0 . 5 , y 5 = 0 . 386637 + 0 . 1 · cos 2 (0 . 386637) = 0 . 472418
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HW_07_Solutions - A Problem 1 Eulers method(a The true...

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