Sr Examen
x(t)=e^(2t)(t^3+3t^2-2t+6) la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2*t / 3 2 \ x*t = E *\t + 3*t - 2*t + 6/
$$t x = e^{2 t} \left(\left(- 2 t + \left(t^{3} + 3 t^{2}\right)\right) + 6\right)$$
Resolución de la ecuación paramétrica
Se da la ecuación con parámetro:
$$t x = \left(t^{3} + 3 t^{2} - 2 t + 6\right) e^{2 t}$$
Коэффициент при x равен
$$t$$
entonces son posibles los casos para t :
$$t < 0$$
$$t = 0$$
Consideremos todos los casos con detalles:
Con
$$t < 0$$
la ecuación será
$$- x - \frac{10}{e^{2}} = 0$$
su solución
$$x = - \frac{10}{e^{2}}$$
Con
$$t = 0$$
la ecuación será
$$-6 = 0$$
su solución
no hay soluciones
$$t x = \left(t^{3} + 3 t^{2} - 2 t + 6\right) e^{2 t}$$
Коэффициент при x равен
$$t$$
entonces son posibles los casos para t :
$$t < 0$$
$$t = 0$$
Consideremos todos los casos con detalles:
Con
$$t < 0$$
la ecuación será
$$- x - \frac{10}{e^{2}} = 0$$
su solución
$$x = - \frac{10}{e^{2}}$$
Con
$$t = 0$$
la ecuación será
$$-6 = 0$$
su solución
no hay soluciones
Gráfica
Suma y producto de raíces
[src]
suma
/ / 2*t\ / 2*t\\ / 2*t\ / 2*t\ |/ / 2 2 \ \ |e | / / 2 2 \ \ |e || / / 2 2 \ \ |e | / / 2 2 \ \ |e | I*|\(3*im(t) + 2*im(t)*re(t))*re(t) + \-2 + re (t) - im (t) + 3*re(t)/*im(t)/*re|----| + \6 + \-2 + re (t) - im (t) + 3*re(t)/*re(t) - (3*im(t) + 2*im(t)*re(t))*im(t)/*im|----|| + \6 + \-2 + re (t) - im (t) + 3*re(t)/*re(t) - (3*im(t) + 2*im(t)*re(t))*im(t)/*re|----| - \(3*im(t) + 2*im(t)*re(t))*re(t) + \-2 + re (t) - im (t) + 3*re(t)/*im(t)/*im|----| \ \ t / \ t // \ t / \ t /
$$i \left(\left(\left(2 \operatorname{re}{\left(t\right)} \operatorname{im}{\left(t\right)} + 3 \operatorname{im}{\left(t\right)}\right) \operatorname{re}{\left(t\right)} + \left(\left(\operatorname{re}{\left(t\right)}\right)^{2} + 3 \operatorname{re}{\left(t\right)} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 2\right) \operatorname{im}{\left(t\right)}\right) \operatorname{re}{\left(\frac{e^{2 t}}{t}\right)} + \left(- \left(2 \operatorname{re}{\left(t\right)} \operatorname{im}{\left(t\right)} + 3 \operatorname{im}{\left(t\right)}\right) \operatorname{im}{\left(t\right)} + \left(\left(\operatorname{re}{\left(t\right)}\right)^{2} + 3 \operatorname{re}{\left(t\right)} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 2\right) \operatorname{re}{\left(t\right)} + 6\right) \operatorname{im}{\left(\frac{e^{2 t}}{t}\right)}\right) - \left(\left(2 \operatorname{re}{\left(t\right)} \operatorname{im}{\left(t\right)} + 3 \operatorname{im}{\left(t\right)}\right) \operatorname{re}{\left(t\right)} + \left(\left(\operatorname{re}{\left(t\right)}\right)^{2} + 3 \operatorname{re}{\left(t\right)} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 2\right) \operatorname{im}{\left(t\right)}\right) \operatorname{im}{\left(\frac{e^{2 t}}{t}\right)} + \left(- \left(2 \operatorname{re}{\left(t\right)} \operatorname{im}{\left(t\right)} + 3 \operatorname{im}{\left(t\right)}\right) \operatorname{im}{\left(t\right)} + \left(\left(\operatorname{re}{\left(t\right)}\right)^{2} + 3 \operatorname{re}{\left(t\right)} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 2\right) \operatorname{re}{\left(t\right)} + 6\right) \operatorname{re}{\left(\frac{e^{2 t}}{t}\right)}$$
=
/ / 2*t\ / 2*t\\ / 2*t\ / 2*t\ |/ / 2 2 \ \ |e | / / 2 2 \ \ |e || / / 2 2 \ \ |e | / / 2 2 \ \ |e | I*|\(3*im(t) + 2*im(t)*re(t))*re(t) + \-2 + re (t) - im (t) + 3*re(t)/*im(t)/*re|----| + \6 + \-2 + re (t) - im (t) + 3*re(t)/*re(t) - (3*im(t) + 2*im(t)*re(t))*im(t)/*im|----|| + \6 + \-2 + re (t) - im (t) + 3*re(t)/*re(t) - (3*im(t) + 2*im(t)*re(t))*im(t)/*re|----| - \(3*im(t) + 2*im(t)*re(t))*re(t) + \-2 + re (t) - im (t) + 3*re(t)/*im(t)/*im|----| \ \ t / \ t // \ t / \ t /
$$i \left(\left(\left(2 \operatorname{re}{\left(t\right)} \operatorname{im}{\left(t\right)} + 3 \operatorname{im}{\left(t\right)}\right) \operatorname{re}{\left(t\right)} + \left(\left(\operatorname{re}{\left(t\right)}\right)^{2} + 3 \operatorname{re}{\left(t\right)} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 2\right) \operatorname{im}{\left(t\right)}\right) \operatorname{re}{\left(\frac{e^{2 t}}{t}\right)} + \left(- \left(2 \operatorname{re}{\left(t\right)} \operatorname{im}{\left(t\right)} + 3 \operatorname{im}{\left(t\right)}\right) \operatorname{im}{\left(t\right)} + \left(\left(\operatorname{re}{\left(t\right)}\right)^{2} + 3 \operatorname{re}{\left(t\right)} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 2\right) \operatorname{re}{\left(t\right)} + 6\right) \operatorname{im}{\left(\frac{e^{2 t}}{t}\right)}\right) - \left(\left(2 \operatorname{re}{\left(t\right)} \operatorname{im}{\left(t\right)} + 3 \operatorname{im}{\left(t\right)}\right) \operatorname{re}{\left(t\right)} + \left(\left(\operatorname{re}{\left(t\right)}\right)^{2} + 3 \operatorname{re}{\left(t\right)} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 2\right) \operatorname{im}{\left(t\right)}\right) \operatorname{im}{\left(\frac{e^{2 t}}{t}\right)} + \left(- \left(2 \operatorname{re}{\left(t\right)} \operatorname{im}{\left(t\right)} + 3 \operatorname{im}{\left(t\right)}\right) \operatorname{im}{\left(t\right)} + \left(\left(\operatorname{re}{\left(t\right)}\right)^{2} + 3 \operatorname{re}{\left(t\right)} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 2\right) \operatorname{re}{\left(t\right)} + 6\right) \operatorname{re}{\left(\frac{e^{2 t}}{t}\right)}$$
producto
/ / 2*t\ / 2*t\\ / 2*t\ / 2*t\ |/ / 2 2 \ \ |e | / / 2 2 \ \ |e || / / 2 2 \ \ |e | / / 2 2 \ \ |e | I*|\(3*im(t) + 2*im(t)*re(t))*re(t) + \-2 + re (t) - im (t) + 3*re(t)/*im(t)/*re|----| + \6 + \-2 + re (t) - im (t) + 3*re(t)/*re(t) - (3*im(t) + 2*im(t)*re(t))*im(t)/*im|----|| + \6 + \-2 + re (t) - im (t) + 3*re(t)/*re(t) - (3*im(t) + 2*im(t)*re(t))*im(t)/*re|----| - \(3*im(t) + 2*im(t)*re(t))*re(t) + \-2 + re (t) - im (t) + 3*re(t)/*im(t)/*im|----| \ \ t / \ t // \ t / \ t /
$$i \left(\left(\left(2 \operatorname{re}{\left(t\right)} \operatorname{im}{\left(t\right)} + 3 \operatorname{im}{\left(t\right)}\right) \operatorname{re}{\left(t\right)} + \left(\left(\operatorname{re}{\left(t\right)}\right)^{2} + 3 \operatorname{re}{\left(t\right)} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 2\right) \operatorname{im}{\left(t\right)}\right) \operatorname{re}{\left(\frac{e^{2 t}}{t}\right)} + \left(- \left(2 \operatorname{re}{\left(t\right)} \operatorname{im}{\left(t\right)} + 3 \operatorname{im}{\left(t\right)}\right) \operatorname{im}{\left(t\right)} + \left(\left(\operatorname{re}{\left(t\right)}\right)^{2} + 3 \operatorname{re}{\left(t\right)} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 2\right) \operatorname{re}{\left(t\right)} + 6\right) \operatorname{im}{\left(\frac{e^{2 t}}{t}\right)}\right) - \left(\left(2 \operatorname{re}{\left(t\right)} \operatorname{im}{\left(t\right)} + 3 \operatorname{im}{\left(t\right)}\right) \operatorname{re}{\left(t\right)} + \left(\left(\operatorname{re}{\left(t\right)}\right)^{2} + 3 \operatorname{re}{\left(t\right)} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 2\right) \operatorname{im}{\left(t\right)}\right) \operatorname{im}{\left(\frac{e^{2 t}}{t}\right)} + \left(- \left(2 \operatorname{re}{\left(t\right)} \operatorname{im}{\left(t\right)} + 3 \operatorname{im}{\left(t\right)}\right) \operatorname{im}{\left(t\right)} + \left(\left(\operatorname{re}{\left(t\right)}\right)^{2} + 3 \operatorname{re}{\left(t\right)} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 2\right) \operatorname{re}{\left(t\right)} + 6\right) \operatorname{re}{\left(\frac{e^{2 t}}{t}\right)}$$
=
/ / 2*t\ / 2*t\\ / 2*t\ / 2*t\ |/ / 2 2 \ 2 \ |e | / 2 2 \ |e || / / 2 2 \ 2 \ |e | / 2 2 \ |e | I*|\6 + \-2 + re (t) - im (t) + 3*re(t)/*re(t) - im (t)*(3 + 2*re(t))/*im|----| + \-2 + re (t) - im (t) + 3*re(t) + (3 + 2*re(t))*re(t)/*im(t)*re|----|| + \6 + \-2 + re (t) - im (t) + 3*re(t)/*re(t) - im (t)*(3 + 2*re(t))/*re|----| - \-2 + re (t) - im (t) + 3*re(t) + (3 + 2*re(t))*re(t)/*im(t)*im|----| \ \ t / \ t // \ t / \ t /
$$i \left(\left(- \left(2 \operatorname{re}{\left(t\right)} + 3\right) \left(\operatorname{im}{\left(t\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(t\right)}\right)^{2} + 3 \operatorname{re}{\left(t\right)} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 2\right) \operatorname{re}{\left(t\right)} + 6\right) \operatorname{im}{\left(\frac{e^{2 t}}{t}\right)} + \left(\left(2 \operatorname{re}{\left(t\right)} + 3\right) \operatorname{re}{\left(t\right)} + \left(\operatorname{re}{\left(t\right)}\right)^{2} + 3 \operatorname{re}{\left(t\right)} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 2\right) \operatorname{re}{\left(\frac{e^{2 t}}{t}\right)} \operatorname{im}{\left(t\right)}\right) + \left(- \left(2 \operatorname{re}{\left(t\right)} + 3\right) \left(\operatorname{im}{\left(t\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(t\right)}\right)^{2} + 3 \operatorname{re}{\left(t\right)} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 2\right) \operatorname{re}{\left(t\right)} + 6\right) \operatorname{re}{\left(\frac{e^{2 t}}{t}\right)} - \left(\left(2 \operatorname{re}{\left(t\right)} + 3\right) \operatorname{re}{\left(t\right)} + \left(\operatorname{re}{\left(t\right)}\right)^{2} + 3 \operatorname{re}{\left(t\right)} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 2\right) \operatorname{im}{\left(t\right)} \operatorname{im}{\left(\frac{e^{2 t}}{t}\right)}$$
i*((6 + (-2 + re(t)^2 - im(t)^2 + 3*re(t))*re(t) - im(t)^2*(3 + 2*re(t)))*im(exp(2*t)/t) + (-2 + re(t)^2 - im(t)^2 + 3*re(t) + (3 + 2*re(t))*re(t))*im(t)*re(exp(2*t)/t)) + (6 + (-2 + re(t)^2 - im(t)^2 + 3*re(t))*re(t) - im(t)^2*(3 + 2*re(t)))*re(exp(2*t)/t) - (-2 + re(t)^2 - im(t)^2 + 3*re(t) + (3 + 2*re(t))*re(t))*im(t)*im(exp(2*t)/t)
Respuesta rápida
[src]
/ / 2*t\ / 2*t\\ / 2*t\ / 2*t\
|/ / 2 2 \ \ |e | / / 2 2 \ \ |e || / / 2 2 \ \ |e | / / 2 2 \ \ |e |
x1 = I*|\(3*im(t) + 2*im(t)*re(t))*re(t) + \-2 + re (t) - im (t) + 3*re(t)/*im(t)/*re|----| + \6 + \-2 + re (t) - im (t) + 3*re(t)/*re(t) - (3*im(t) + 2*im(t)*re(t))*im(t)/*im|----|| + \6 + \-2 + re (t) - im (t) + 3*re(t)/*re(t) - (3*im(t) + 2*im(t)*re(t))*im(t)/*re|----| - \(3*im(t) + 2*im(t)*re(t))*re(t) + \-2 + re (t) - im (t) + 3*re(t)/*im(t)/*im|----|
\ \ t / \ t // \ t / \ t /
$$x_{1} = i \left(\left(\left(2 \operatorname{re}{\left(t\right)} \operatorname{im}{\left(t\right)} + 3 \operatorname{im}{\left(t\right)}\right) \operatorname{re}{\left(t\right)} + \left(\left(\operatorname{re}{\left(t\right)}\right)^{2} + 3 \operatorname{re}{\left(t\right)} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 2\right) \operatorname{im}{\left(t\right)}\right) \operatorname{re}{\left(\frac{e^{2 t}}{t}\right)} + \left(- \left(2 \operatorname{re}{\left(t\right)} \operatorname{im}{\left(t\right)} + 3 \operatorname{im}{\left(t\right)}\right) \operatorname{im}{\left(t\right)} + \left(\left(\operatorname{re}{\left(t\right)}\right)^{2} + 3 \operatorname{re}{\left(t\right)} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 2\right) \operatorname{re}{\left(t\right)} + 6\right) \operatorname{im}{\left(\frac{e^{2 t}}{t}\right)}\right) - \left(\left(2 \operatorname{re}{\left(t\right)} \operatorname{im}{\left(t\right)} + 3 \operatorname{im}{\left(t\right)}\right) \operatorname{re}{\left(t\right)} + \left(\left(\operatorname{re}{\left(t\right)}\right)^{2} + 3 \operatorname{re}{\left(t\right)} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 2\right) \operatorname{im}{\left(t\right)}\right) \operatorname{im}{\left(\frac{e^{2 t}}{t}\right)} + \left(- \left(2 \operatorname{re}{\left(t\right)} \operatorname{im}{\left(t\right)} + 3 \operatorname{im}{\left(t\right)}\right) \operatorname{im}{\left(t\right)} + \left(\left(\operatorname{re}{\left(t\right)}\right)^{2} + 3 \operatorname{re}{\left(t\right)} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 2\right) \operatorname{re}{\left(t\right)} + 6\right) \operatorname{re}{\left(\frac{e^{2 t}}{t}\right)}$$
x1 = i*(((2*re(t)*im(t) + 3*im(t))*re(t) + (re(t)^2 + 3*re(t) - im(t)^2 - 2)*im(t))*re(exp(2*t)/t) + (-(2*re(t)*im(t) + 3*im(t))*im(t) + (re(t)^2 + 3*re(t) - im(t)^2 - 2)*re(t) + 6)*im(exp(2*t)/t)) - ((2*re(t)*im(t) + 3*im(t))*re(t) + (re(t)^2 + 3*re(t) - im(t)^2 - 2)*im(t))*im(exp(2*t)/t) + (-(2*re(t)*im(t) + 3*im(t))*im(t) + (re(t)^2 + 3*re(t) - im(t)^2 - 2)*re(t) + 6)*re(exp(2*t)/t)