Calculadora del método de Euler

La calculadora encontrará la solución aproximada de la ecuación diferencial de primer orden utilizando el método de Euler, 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 $$$y^{\prime } = t y$$$, cuando $$$y{\left(0 \right)} = 3$$$, $$$h = \frac{1}{5}$$$ usando el método de Euler.

Solución

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

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

Paso 1

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

$$$y{\left(\frac{1}{5} \right)} = y{\left(t_{1} \right)} = y_{1} = y_{0} + h\cdot f{\left(t_{0},y_{0} \right)} = 3 + h\cdot f{\left(0,3 \right)} = 3 + \frac{1}{5} \cdot 0 = 3$$$

Paso 2

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

$$$y{\left(\frac{2}{5} \right)} = y{\left(t_{2} \right)} = y_{2} = y_{1} + h\cdot f{\left(t_{1},y_{1} \right)} = 3 + h\cdot f{\left(\frac{1}{5},3 \right)} = 3 + \frac{1}{5} \cdot 0.6 = 3.12$$$

Paso 3

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

$$$y{\left(\frac{3}{5} \right)} = y{\left(t_{3} \right)} = y_{3} = y_{2} + h\cdot f{\left(t_{2},y_{2} \right)} = 3.12 + h\cdot f{\left(\frac{2}{5},3.12 \right)} = 3.12 + \frac{1}{5} \cdot 1.248 = 3.3696$$$

Paso 4

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

$$$y{\left(\frac{4}{5} \right)} = y{\left(t_{4} \right)} = y_{4} = y_{3} + h\cdot f{\left(t_{3},y_{3} \right)} = 3.3696 + h\cdot f{\left(\frac{3}{5},3.3696 \right)} = 3.3696 + \frac{1}{5} \cdot 2.02176 = 3.773952$$$

Paso 5

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

$$$y{\left(1 \right)} = y{\left(t_{5} \right)} = y_{5} = y_{4} + h\cdot f{\left(t_{4},y_{4} \right)} = 3.773952 + h\cdot f{\left(\frac{4}{5},3.773952 \right)} = 3.773952 + \frac{1}{5} \cdot 3.0191616 = 4.37778432$$$

Respuesta

$$$y{\left(1 \right)}\approx 4.37778432$$$A