Euler_methods_predictor_corrector - Euler Methods November...

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Euler Methods November 5, 2016 1 Analytical solution First of all, let us find out the analytical solution to this IVP. dy dx = 2 xy (1) with the Initial condition y (1) = 1 (2) Eq, (1) can be integrated using separation of variables, dy y = 2 xdx (3) integrating, gives us Z dy y = Z 2 xdx (4) lny = x 2 + C (5) C is a constant of integration, Eq. (4) can be written as y ( x ) = Ce x 2 C = 1 e (6) Using the initial condition Eq. (2), gives y (1) = Ce = 1 (7) then, the analytical solution of IVP (1)-(2) is y ( x ) = e x 2 - 1 (8) 1
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2 Explicit method dy dx = F ( x, y ) = 2 xy (9) solution, y n +1 - y n Δ x = F ( x n , y n ) = 2 x n y n (10) therefore y n +1 = y n + F ( x n , y n x = y n + 2 x n y n Δ x (11) 3 Implicit method dy dx = F ( x, y ) = 2 xy (12) y n +1 - y n Δ x = F ( x n +1 , y n +1
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