Ejercicio con Método de Euler (h = 0.1)
Ejercicio 9: Método de Euler (3 iteraciones, (h = 0.1))
Dada la EDO:
\[y' = x + y, \quad y(0) = 1\]El método de Euler de primer orden utiliza la fórmula
\[y_{n+1} = y_n + h\,f(x_n, y_n),\]donde
\[f(x,y) = x + y.\]Paso 0 → 1 ((x = 0) a (x = 0.1))
\[\begin{aligned} x_0 &= 0, \\ y_0 &= 1, \\ f_0 &= 0 + 1 = 1, \\ y_1 &= y_0 + h\,f_0 = 1 + 0.1(1) = 1.1. \end{aligned}\]Paso 1 → 2 ((x = 0.1) a (x = 0.2))
\[\begin{aligned} x_1 &= 0.1, \\ y_1 &= 1.1, \\ f_1 &= 0.1 + 1.1 = 1.2, \\ y_2 &= y_1 + h\,f_1 = 1.1 + 0.1(1.2) = 1.22. \end{aligned}\]Paso 2 → 3 ((x = 0.2) a (x = 0.3))
\[\begin{aligned} x_2 &= 0.2, \\ y_2 &= 1.22, \\ f_2 &= 0.2 + 1.22 = 1.42, \\ y_3 &= y_2 + h\,f_2 = 1.22 + 0.1(1.42) = 1.362. \end{aligned}\]\[y(0.3) \approx 1.362\]