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- La ecuación:
-
Ecuación x^2-4x+5=0
-
Ecuación x^2+3x+2=0
-
Ecuación x^2+3x=4
-
Ecuación -x^2+5*x-6=0
- Expresar {x} en función de y en la ecuación:
- 10*x-2*y=0
- 1*x+4*y=20
- 3*x-7*y=13
- 13*x-15*y=-6
-
Expresiones idénticas
- Xy-y=xtany/x
- Xy menos y es igual a x tangente de y dividir por x
- Xy-y=xtany dividir por x
-
Expresiones semejantes
- Xy+y=xtany/x
Xy-y=xtany/x la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Solución detallada
Tenemos la ecuación
$$x y - y = \frac{x \tan{\left(y \right)}}{x}$$
cambiamos
$$x y - y - \tan{\left(y \right)} - 1 = 0$$
$$x y - y - 1 - \frac{x \tan{\left(y \right)}}{x} = 0$$
Sustituimos
$$w = \tan{\left(y \right)}$$
Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
$$- w + x y - y = 1$$
Move the summands with the other variables
del miembro izquierdo al derecho, obtenemos:
$$- w + x y = y + 1$$
Dividamos ambos miembros de la ecuación en (-w + x*y)/w
Obtenemos la respuesta: w = -1 - y + x*y
hacemos cambio inverso
$$\tan{\left(y \right)} = w$$
sustituimos w:
$$x y - y = \frac{x \tan{\left(y \right)}}{x}$$
cambiamos
$$x y - y - \tan{\left(y \right)} - 1 = 0$$
$$x y - y - 1 - \frac{x \tan{\left(y \right)}}{x} = 0$$
Sustituimos
$$w = \tan{\left(y \right)}$$
Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
$$- w + x y - y = 1$$
Move the summands with the other variables
del miembro izquierdo al derecho, obtenemos:
$$- w + x y = y + 1$$
Dividamos ambos miembros de la ecuación en (-w + x*y)/w
w = 1 + y / ((-w + x*y)/w)
Obtenemos la respuesta: w = -1 - y + x*y
hacemos cambio inverso
$$\tan{\left(y \right)} = w$$
sustituimos w:
Resolución de la ecuación paramétrica
Se da la ecuación con parámetro:
$$x y - y = \tan{\left(y \right)}$$
Коэффициент при x равен
$$y$$
entonces son posibles los casos para y :
$$y < 0$$
$$y = 0$$
Consideremos todos los casos con detalles:
Con
$$y < 0$$
la ecuación será
$$- x + 1 + \tan{\left(1 \right)} = 0$$
su solución
$$x = 1 + \tan{\left(1 \right)}$$
Con
$$y = 0$$
la ecuación será
$$0 = 0$$
su solución
cualquiera x
$$x y - y = \tan{\left(y \right)}$$
Коэффициент при x равен
$$y$$
entonces son posibles los casos para y :
$$y < 0$$
$$y = 0$$
Consideremos todos los casos con detalles:
Con
$$y < 0$$
la ecuación será
$$- x + 1 + \tan{\left(1 \right)} = 0$$
su solución
$$x = 1 + \tan{\left(1 \right)}$$
Con
$$y = 0$$
la ecuación será
$$0 = 0$$
su solución
cualquiera x
Gráfica
Respuesta rápida
[src]
// sinh(2*im(y)) \ / sin(2*re(y)) \ \ / sin(2*re(y)) \ / sinh(2*im(y)) \
||---------------------------- + im(y)|*re(y) |---------------------------- + re(y)|*im(y)| |---------------------------- + re(y)|*re(y) |---------------------------- + im(y)|*im(y)
|\cos(2*re(y)) + cosh(2*im(y)) / \cos(2*re(y)) + cosh(2*im(y)) / | \cos(2*re(y)) + cosh(2*im(y)) / \cos(2*re(y)) + cosh(2*im(y)) /
x1 = I*|-------------------------------------------- - --------------------------------------------| + -------------------------------------------- + --------------------------------------------
| 2 2 2 2 | 2 2 2 2
\ im (y) + re (y) im (y) + re (y) / im (y) + re (y) im (y) + re (y)
$$x_{1} = i \left(- \frac{\left(\operatorname{re}{\left(y\right)} + \frac{\sin{\left(2 \operatorname{re}{\left(y\right)} \right)}}{\cos{\left(2 \operatorname{re}{\left(y\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(y\right)} \right)}}\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(y\right)} + \frac{\sinh{\left(2 \operatorname{im}{\left(y\right)} \right)}}{\cos{\left(2 \operatorname{re}{\left(y\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(y\right)} \right)}}\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right) + \frac{\left(\operatorname{re}{\left(y\right)} + \frac{\sin{\left(2 \operatorname{re}{\left(y\right)} \right)}}{\cos{\left(2 \operatorname{re}{\left(y\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(y\right)} \right)}}\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(y\right)} + \frac{\sinh{\left(2 \operatorname{im}{\left(y\right)} \right)}}{\cos{\left(2 \operatorname{re}{\left(y\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(y\right)} \right)}}\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}$$
x1 = i*(-(re(y) + sin(2*re(y))/(cos(2*re(y)) + cosh(2*im(y))))*im(y)/(re(y)^2 + im(y)^2) + (im(y) + sinh(2*im(y))/(cos(2*re(y)) + cosh(2*im(y))))*re(y)/(re(y)^2 + im(y)^2)) + (re(y) + sin(2*re(y))/(cos(2*re(y)) + cosh(2*im(y))))*re(y)/(re(y)^2 + im(y)^2) + (im(y) + sinh(2*im(y))/(cos(2*re(y)) + cosh(2*im(y))))*im(y)/(re(y)^2 + im(y)^2)
Suma y producto de raíces
[src]
suma
// sinh(2*im(y)) \ / sin(2*re(y)) \ \ / sin(2*re(y)) \ / sinh(2*im(y)) \ ||---------------------------- + im(y)|*re(y) |---------------------------- + re(y)|*im(y)| |---------------------------- + re(y)|*re(y) |---------------------------- + im(y)|*im(y) |\cos(2*re(y)) + cosh(2*im(y)) / \cos(2*re(y)) + cosh(2*im(y)) / | \cos(2*re(y)) + cosh(2*im(y)) / \cos(2*re(y)) + cosh(2*im(y)) / I*|-------------------------------------------- - --------------------------------------------| + -------------------------------------------- + -------------------------------------------- | 2 2 2 2 | 2 2 2 2 \ im (y) + re (y) im (y) + re (y) / im (y) + re (y) im (y) + re (y)
$$i \left(- \frac{\left(\operatorname{re}{\left(y\right)} + \frac{\sin{\left(2 \operatorname{re}{\left(y\right)} \right)}}{\cos{\left(2 \operatorname{re}{\left(y\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(y\right)} \right)}}\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(y\right)} + \frac{\sinh{\left(2 \operatorname{im}{\left(y\right)} \right)}}{\cos{\left(2 \operatorname{re}{\left(y\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(y\right)} \right)}}\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right) + \frac{\left(\operatorname{re}{\left(y\right)} + \frac{\sin{\left(2 \operatorname{re}{\left(y\right)} \right)}}{\cos{\left(2 \operatorname{re}{\left(y\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(y\right)} \right)}}\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(y\right)} + \frac{\sinh{\left(2 \operatorname{im}{\left(y\right)} \right)}}{\cos{\left(2 \operatorname{re}{\left(y\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(y\right)} \right)}}\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}$$
=
// sinh(2*im(y)) \ / sin(2*re(y)) \ \ / sin(2*re(y)) \ / sinh(2*im(y)) \ ||---------------------------- + im(y)|*re(y) |---------------------------- + re(y)|*im(y)| |---------------------------- + re(y)|*re(y) |---------------------------- + im(y)|*im(y) |\cos(2*re(y)) + cosh(2*im(y)) / \cos(2*re(y)) + cosh(2*im(y)) / | \cos(2*re(y)) + cosh(2*im(y)) / \cos(2*re(y)) + cosh(2*im(y)) / I*|-------------------------------------------- - --------------------------------------------| + -------------------------------------------- + -------------------------------------------- | 2 2 2 2 | 2 2 2 2 \ im (y) + re (y) im (y) + re (y) / im (y) + re (y) im (y) + re (y)
$$i \left(- \frac{\left(\operatorname{re}{\left(y\right)} + \frac{\sin{\left(2 \operatorname{re}{\left(y\right)} \right)}}{\cos{\left(2 \operatorname{re}{\left(y\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(y\right)} \right)}}\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(y\right)} + \frac{\sinh{\left(2 \operatorname{im}{\left(y\right)} \right)}}{\cos{\left(2 \operatorname{re}{\left(y\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(y\right)} \right)}}\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right) + \frac{\left(\operatorname{re}{\left(y\right)} + \frac{\sin{\left(2 \operatorname{re}{\left(y\right)} \right)}}{\cos{\left(2 \operatorname{re}{\left(y\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(y\right)} \right)}}\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(y\right)} + \frac{\sinh{\left(2 \operatorname{im}{\left(y\right)} \right)}}{\cos{\left(2 \operatorname{re}{\left(y\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(y\right)} \right)}}\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}$$
producto
// sinh(2*im(y)) \ / sin(2*re(y)) \ \ / sin(2*re(y)) \ / sinh(2*im(y)) \ ||---------------------------- + im(y)|*re(y) |---------------------------- + re(y)|*im(y)| |---------------------------- + re(y)|*re(y) |---------------------------- + im(y)|*im(y) |\cos(2*re(y)) + cosh(2*im(y)) / \cos(2*re(y)) + cosh(2*im(y)) / | \cos(2*re(y)) + cosh(2*im(y)) / \cos(2*re(y)) + cosh(2*im(y)) / I*|-------------------------------------------- - --------------------------------------------| + -------------------------------------------- + -------------------------------------------- | 2 2 2 2 | 2 2 2 2 \ im (y) + re (y) im (y) + re (y) / im (y) + re (y) im (y) + re (y)
$$i \left(- \frac{\left(\operatorname{re}{\left(y\right)} + \frac{\sin{\left(2 \operatorname{re}{\left(y\right)} \right)}}{\cos{\left(2 \operatorname{re}{\left(y\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(y\right)} \right)}}\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(y\right)} + \frac{\sinh{\left(2 \operatorname{im}{\left(y\right)} \right)}}{\cos{\left(2 \operatorname{re}{\left(y\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(y\right)} \right)}}\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right) + \frac{\left(\operatorname{re}{\left(y\right)} + \frac{\sin{\left(2 \operatorname{re}{\left(y\right)} \right)}}{\cos{\left(2 \operatorname{re}{\left(y\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(y\right)} \right)}}\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(y\right)} + \frac{\sinh{\left(2 \operatorname{im}{\left(y\right)} \right)}}{\cos{\left(2 \operatorname{re}{\left(y\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(y\right)} \right)}}\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}$$
=
I*(re(y)*sinh(2*im(y)) - im(y)*sin(2*re(y))) + ((cos(2*re(y)) + cosh(2*im(y)))*im(y) + sinh(2*im(y)))*im(y) + ((cos(2*re(y)) + cosh(2*im(y)))*re(y) + sin(2*re(y)))*re(y)
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------
/ 2 2 \
\im (y) + re (y)/*(cos(2*re(y)) + cosh(2*im(y)))
$$\frac{\left(\left(\cos{\left(2 \operatorname{re}{\left(y\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(y\right)} \right)}\right) \operatorname{re}{\left(y\right)} + \sin{\left(2 \operatorname{re}{\left(y\right)} \right)}\right) \operatorname{re}{\left(y\right)} + \left(\left(\cos{\left(2 \operatorname{re}{\left(y\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(y\right)} \right)}\right) \operatorname{im}{\left(y\right)} + \sinh{\left(2 \operatorname{im}{\left(y\right)} \right)}\right) \operatorname{im}{\left(y\right)} + i \left(- \sin{\left(2 \operatorname{re}{\left(y\right)} \right)} \operatorname{im}{\left(y\right)} + \sinh{\left(2 \operatorname{im}{\left(y\right)} \right)} \operatorname{re}{\left(y\right)}\right)}{\left(\cos{\left(2 \operatorname{re}{\left(y\right)} \right)} + \cosh{\left(2 \operatorname{im}{\left(y\right)} \right)}\right) \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}\right)}$$
(i*(re(y)*sinh(2*im(y)) - im(y)*sin(2*re(y))) + ((cos(2*re(y)) + cosh(2*im(y)))*im(y) + sinh(2*im(y)))*im(y) + ((cos(2*re(y)) + cosh(2*im(y)))*re(y) + sin(2*re(y)))*re(y))/((im(y)^2 + re(y)^2)*(cos(2*re(y)) + cosh(2*im(y))))