Calculadora de método de Euler (Heun) mejorada

La calculadora encontrará la solución aproximada de la ecuación diferencial de primer orden utilizando el método mejorado de Euler (Heun), con los pasos que se muestran.

Si la calculadora no calculó algo o si ha identificado un error, o si tiene una sugerencia / comentario, escríbalo en los comentarios a continuación.

Tu aportación

Encuentre $$$y{\left(1 \right)}$$$ para $$$y^{\prime } = 3 t + y$$$, cuando $$$y{\left(0 \right)} = 7$$$, $$$h = \frac{1}{5}$$$ usando el método de Euler mejorado.

Solución

El método de Euler mejorado establece que $$$y_{n+1} = y_{n} + \frac{h}{2} \left(f{\left(t_{n},y_{n} \right)} + f{\left(t_{n+1},\tilde{y}_{n+1} \right)}\right)$$$, donde $$$\tilde{y}_{n+1} = y_{n} + h\cdot f{\left(t_{n},y_{n} \right)}$$$ y $$$t_{n+1} = t_{n} + h$$$.

Tenemos eso $$$h = \frac{1}{5}$$$, $$$t_{0} = 0$$$, $$$y_{0} = 7$$$ $$$f{\left(t,y \right)} = 3 t + y$$$.

Paso 1

$$$t_{1} = t_{0} + h = 0 + \frac{1}{5} = \frac{1}{5}$$$

$$$\tilde{y}{\left(\frac{1}{5} \right)} = \tilde{y}{\left(t_{1} \right)} = y_{1} = y_{0} + h\cdot f{\left(t_{0},y_{0} \right)} = 7 + h\cdot f{\left(0,7 \right)} = 7 + \frac{1}{5} \cdot 7 = 8.4$$$

$$$y{\left(\frac{1}{5} \right)} = y{\left(t_{1} \right)} = y_{1} = y_{0} + \frac{h}{2} \left(f{\left(t_{0},y_{0} \right)} + f{\left(t_{1},\tilde{y}_{1} \right)}\right) = 7 + \frac{h}{2} \left(f{\left(0,7 \right)} + f{\left(\frac{1}{5},8.4 \right)}\right) = 7 + \frac{\frac{1}{5}}{2} \left(7 + 9\right) = 8.6$$$

Paso 2

$$$t_{2} = t_{1} + h = \frac{1}{5} + \frac{1}{5} = \frac{2}{5}$$$

$$$\tilde{y}{\left(\frac{2}{5} \right)} = \tilde{y}{\left(t_{2} \right)} = y_{2} = y_{1} + h\cdot f{\left(t_{1},y_{1} \right)} = 8.6 + h\cdot f{\left(\frac{1}{5},8.6 \right)} = 8.6 + \frac{1}{5} \cdot 9.199999999999999 = 10.44$$$

$$$y{\left(\frac{2}{5} \right)} = y{\left(t_{2} \right)} = y_{2} = y_{1} + \frac{h}{2} \left(f{\left(t_{1},y_{1} \right)} + f{\left(t_{2},\tilde{y}_{2} \right)}\right) = 8.6 + \frac{h}{2} \left(f{\left(\frac{1}{5},8.6 \right)} + f{\left(\frac{2}{5},10.44 \right)}\right) = 8.6 + \frac{\frac{1}{5}}{2} \left(9.199999999999999 + 11.639999999999999\right) = 10.684$$$

Paso 3

$$$t_{3} = t_{2} + h = \frac{2}{5} + \frac{1}{5} = \frac{3}{5}$$$

$$$\tilde{y}{\left(\frac{3}{5} \right)} = \tilde{y}{\left(t_{3} \right)} = y_{3} = y_{2} + h\cdot f{\left(t_{2},y_{2} \right)} = 10.684 + h\cdot f{\left(\frac{2}{5},10.684 \right)} = 10.684 + \frac{1}{5} \cdot 11.884 = 13.0608$$$

$$$y{\left(\frac{3}{5} \right)} = y{\left(t_{3} \right)} = y_{3} = y_{2} + \frac{h}{2} \left(f{\left(t_{2},y_{2} \right)} + f{\left(t_{3},\tilde{y}_{3} \right)}\right) = 10.684 + \frac{h}{2} \left(f{\left(\frac{2}{5},10.684 \right)} + f{\left(\frac{3}{5},13.0608 \right)}\right) = 10.684 + \frac{\frac{1}{5}}{2} \left(11.884 + 14.8608\right) = 13.35848$$$

Paso 4

$$$t_{4} = t_{3} + h = \frac{3}{5} + \frac{1}{5} = \frac{4}{5}$$$

$$$\tilde{y}{\left(\frac{4}{5} \right)} = \tilde{y}{\left(t_{4} \right)} = y_{4} = y_{3} + h\cdot f{\left(t_{3},y_{3} \right)} = 13.35848 + h\cdot f{\left(\frac{3}{5},13.35848 \right)} = 13.35848 + \frac{1}{5} \cdot 15.15848 = 16.390176$$$

$$$y{\left(\frac{4}{5} \right)} = y{\left(t_{4} \right)} = y_{4} = y_{3} + \frac{h}{2} \left(f{\left(t_{3},y_{3} \right)} + f{\left(t_{4},\tilde{y}_{4} \right)}\right) = 13.35848 + \frac{h}{2} \left(f{\left(\frac{3}{5},13.35848 \right)} + f{\left(\frac{4}{5},16.390176 \right)}\right) = 13.35848 + \frac{\frac{1}{5}}{2} \left(15.15848 + 18.790176\right) = 16.7533456$$$

Paso 5

$$$t_{5} = t_{4} + h = \frac{4}{5} + \frac{1}{5} = 1$$$

$$$\tilde{y}{\left(1 \right)} = \tilde{y}{\left(t_{5} \right)} = y_{5} = y_{4} + h\cdot f{\left(t_{4},y_{4} \right)} = 16.7533456 + h\cdot f{\left(\frac{4}{5},16.7533456 \right)} = 16.7533456 + \frac{1}{5} \cdot 19.1533456 = 20.58401472$$$

$$$y{\left(1 \right)} = y{\left(t_{5} \right)} = y_{5} = y_{4} + \frac{h}{2} \left(f{\left(t_{4},y_{4} \right)} + f{\left(t_{5},\tilde{y}_{5} \right)}\right) = 16.7533456 + \frac{h}{2} \left(f{\left(\frac{4}{5},16.7533456 \right)} + f{\left(1,20.58401472 \right)}\right) = 16.7533456 + \frac{\frac{1}{5}}{2} \left(19.1533456 + 23.58401472\right) = 21.027081632$$$

Respuesta

$$$y{\left(1 \right)}\approx 21.027081632$$$A