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y=2sin(x)-y*cos(x)+C la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
y = 2*sin(x) - y*cos(x) + c
$$y = c + \left(- y \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right)$$
Simplificar
Resolución de la ecuación paramétrica
Se da la ecuación con parámetro:
$$y = c - y \cos{\left(x \right)} + 2 \sin{\left(x \right)}$$
Коэффициент при y равен
$$\cos{\left(x \right)} + 1$$
entonces son posibles los casos para x :
$$x < \pi$$
$$x = \pi$$
Consideremos todos los casos con detalles:
Con
$$x < \pi$$
la ecuación será
$$- c - y \cos{\left(1 \right)} + y - 2 \sin{\left(1 \right)} = 0$$
su solución
$$y = \frac{c + 2 \sin{\left(1 \right)}}{1 - \cos{\left(1 \right)}}$$
Con
$$x = \pi$$
la ecuación será
$$- c = 0$$
su solución
Gráfica
Suma y producto de raíces [src] 
suma
  /(1 + cos(re(x))*cosh(im(x)))*(2*cos(re(x))*sinh(im(x)) + im(c))   (2*cosh(im(x))*sin(re(x)) + re(c))*sin(re(x))*sinh(im(x))\   (1 + cos(re(x))*cosh(im(x)))*(2*cosh(im(x))*sin(re(x)) + re(c))   (2*cos(re(x))*sinh(im(x)) + im(c))*sin(re(x))*sinh(im(x))
I*|--------------------------------------------------------------- + ---------------------------------------------------------| + --------------------------------------------------------------- - ---------------------------------------------------------
  |                                2      2            2                                          2      2            2       |                                   2      2            2                                          2      2            2       
  \    (1 + cos(re(x))*cosh(im(x)))  + sin (re(x))*sinh (im(x))       (1 + cos(re(x))*cosh(im(x)))  + sin (re(x))*sinh (im(x))/       (1 + cos(re(x))*cosh(im(x)))  + sin (re(x))*sinh (im(x))       (1 + cos(re(x))*cosh(im(x)))  + sin (re(x))*sinh (im(x))
$$\frac{\left(2 \sin{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} + \operatorname{re}{\left(c\right)}\right) \left(\cos{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} + 1\right)}{\left(\cos{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} + 1\right)^{2} + \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)}} - \frac{\left(2 \cos{\left(\operatorname{re}{\left(x\right)} \right)} \sinh{\left(\operatorname{im}{\left(x\right)} \right)} + \operatorname{im}{\left(c\right)}\right) \sin{\left(\operatorname{re}{\left(x\right)} \right)} \sinh{\left(\operatorname{im}{\left(x\right)} \right)}}{\left(\cos{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} + 1\right)^{2} + \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)}} + i \left(\frac{\left(2 \sin{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} + \operatorname{re}{\left(c\right)}\right) \sin{\left(\operatorname{re}{\left(x\right)} \right)} \sinh{\left(\operatorname{im}{\left(x\right)} \right)}}{\left(\cos{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} + 1\right)^{2} + \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)}} + \frac{\left(2 \cos{\left(\operatorname{re}{\left(x\right)} \right)} \sinh{\left(\operatorname{im}{\left(x\right)} \right)} + \operatorname{im}{\left(c\right)}\right) \left(\cos{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} + 1\right)}{\left(\cos{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} + 1\right)^{2} + \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)}}\right)$$ 
=
  /(1 + cos(re(x))*cosh(im(x)))*(2*cos(re(x))*sinh(im(x)) + im(c))   (2*cosh(im(x))*sin(re(x)) + re(c))*sin(re(x))*sinh(im(x))\   (1 + cos(re(x))*cosh(im(x)))*(2*cosh(im(x))*sin(re(x)) + re(c))   (2*cos(re(x))*sinh(im(x)) + im(c))*sin(re(x))*sinh(im(x))
I*|--------------------------------------------------------------- + ---------------------------------------------------------| + --------------------------------------------------------------- - ---------------------------------------------------------
  |                                2      2            2                                          2      2            2       |                                   2      2            2                                          2      2            2       
  \    (1 + cos(re(x))*cosh(im(x)))  + sin (re(x))*sinh (im(x))       (1 + cos(re(x))*cosh(im(x)))  + sin (re(x))*sinh (im(x))/       (1 + cos(re(x))*cosh(im(x)))  + sin (re(x))*sinh (im(x))       (1 + cos(re(x))*cosh(im(x)))  + sin (re(x))*sinh (im(x))
$$\frac{\left(2 \sin{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} + \operatorname{re}{\left(c\right)}\right) \left(\cos{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} + 1\right)}{\left(\cos{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} + 1\right)^{2} + \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)}} - \frac{\left(2 \cos{\left(\operatorname{re}{\left(x\right)} \right)} \sinh{\left(\operatorname{im}{\left(x\right)} \right)} + \operatorname{im}{\left(c\right)}\right) \sin{\left(\operatorname{re}{\left(x\right)} \right)} \sinh{\left(\operatorname{im}{\left(x\right)} \right)}}{\left(\cos{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} + 1\right)^{2} + \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)}} + i \left(\frac{\left(2 \sin{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} + \operatorname{re}{\left(c\right)}\right) \sin{\left(\operatorname{re}{\left(x\right)} \right)} \sinh{\left(\operatorname{im}{\left(x\right)} \right)}}{\left(\cos{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} + 1\right)^{2} + \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)}} + \frac{\left(2 \cos{\left(\operatorname{re}{\left(x\right)} \right)} \sinh{\left(\operatorname{im}{\left(x\right)} \right)} + \operatorname{im}{\left(c\right)}\right) \left(\cos{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} + 1\right)}{\left(\cos{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} + 1\right)^{2} + \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)}}\right)$$ 
producto
  /(1 + cos(re(x))*cosh(im(x)))*(2*cos(re(x))*sinh(im(x)) + im(c))   (2*cosh(im(x))*sin(re(x)) + re(c))*sin(re(x))*sinh(im(x))\   (1 + cos(re(x))*cosh(im(x)))*(2*cosh(im(x))*sin(re(x)) + re(c))   (2*cos(re(x))*sinh(im(x)) + im(c))*sin(re(x))*sinh(im(x))
I*|--------------------------------------------------------------- + ---------------------------------------------------------| + --------------------------------------------------------------- - ---------------------------------------------------------
  |                                2      2            2                                          2      2            2       |                                   2      2            2                                          2      2            2       
  \    (1 + cos(re(x))*cosh(im(x)))  + sin (re(x))*sinh (im(x))       (1 + cos(re(x))*cosh(im(x)))  + sin (re(x))*sinh (im(x))/       (1 + cos(re(x))*cosh(im(x)))  + sin (re(x))*sinh (im(x))       (1 + cos(re(x))*cosh(im(x)))  + sin (re(x))*sinh (im(x))
$$\frac{\left(2 \sin{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} + \operatorname{re}{\left(c\right)}\right) \left(\cos{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} + 1\right)}{\left(\cos{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} + 1\right)^{2} + \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)}} - \frac{\left(2 \cos{\left(\operatorname{re}{\left(x\right)} \right)} \sinh{\left(\operatorname{im}{\left(x\right)} \right)} + \operatorname{im}{\left(c\right)}\right) \sin{\left(\operatorname{re}{\left(x\right)} \right)} \sinh{\left(\operatorname{im}{\left(x\right)} \right)}}{\left(\cos{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} + 1\right)^{2} + \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)}} + i \left(\frac{\left(2 \sin{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} + \operatorname{re}{\left(c\right)}\right) \sin{\left(\operatorname{re}{\left(x\right)} \right)} \sinh{\left(\operatorname{im}{\left(x\right)} \right)}}{\left(\cos{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} + 1\right)^{2} + \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)}} + \frac{\left(2 \cos{\left(\operatorname{re}{\left(x\right)} \right)} \sinh{\left(\operatorname{im}{\left(x\right)} \right)} + \operatorname{im}{\left(c\right)}\right) \left(\cos{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} + 1\right)}{\left(\cos{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} + 1\right)^{2} + \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)}}\right)$$ 
=
2*(I*(2*cos(re(x))*sinh(im(x)) + cos(re(x))*cosh(im(x))*im(c) + re(c)*sin(re(x))*sinh(im(x)) + im(c) + sinh(2*im(x))) + 2*cos(re(x))*sin(re(x)) + 2*cosh(im(x))*sin(re(x)) + cos(re(x))*cosh(im(x))*re(c) - im(c)*sin(re(x))*sinh(im(x)) + re(c))
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                        2             2             2                           2                                                                                                                
                                                                 2 + cos (re(x)) - sin (re(x)) + cos (re(x))*cosh(2*im(x)) + sin (re(x))*cosh(2*im(x)) + 4*cos(re(x))*cosh(im(x))
$$\frac{2 \left(i \left(\sin{\left(\operatorname{re}{\left(x\right)} \right)} \sinh{\left(\operatorname{im}{\left(x\right)} \right)} \operatorname{re}{\left(c\right)} + 2 \cos{\left(\operatorname{re}{\left(x\right)} \right)} \sinh{\left(\operatorname{im}{\left(x\right)} \right)} + \cos{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} \operatorname{im}{\left(c\right)} + \sinh{\left(2 \operatorname{im}{\left(x\right)} \right)} + \operatorname{im}{\left(c\right)}\right) + 2 \sin{\left(\operatorname{re}{\left(x\right)} \right)} \cos{\left(\operatorname{re}{\left(x\right)} \right)} - \sin{\left(\operatorname{re}{\left(x\right)} \right)} \sinh{\left(\operatorname{im}{\left(x\right)} \right)} \operatorname{im}{\left(c\right)} + 2 \sin{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} + \cos{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} \operatorname{re}{\left(c\right)} + \operatorname{re}{\left(c\right)}\right)}{\sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)} - \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} + \cos^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)} + \cos^{2}{\left(\operatorname{re}{\left(x\right)} \right)} + 4 \cos{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} + 2}$$ 
2*(i*(2*cos(re(x))*sinh(im(x)) + cos(re(x))*cosh(im(x))*im(c) + re(c)*sin(re(x))*sinh(im(x)) + im(c) + sinh(2*im(x))) + 2*cos(re(x))*sin(re(x)) + 2*cosh(im(x))*sin(re(x)) + cos(re(x))*cosh(im(x))*re(c) - im(c)*sin(re(x))*sinh(im(x)) + re(c))/(2 + cos(re(x))^2 - sin(re(x))^2 + cos(re(x))^2*cosh(2*im(x)) + sin(re(x))^2*cosh(2*im(x)) + 4*cos(re(x))*cosh(im(x)))
Respuesta rápida [src] 
       /(1 + cos(re(x))*cosh(im(x)))*(2*cos(re(x))*sinh(im(x)) + im(c))   (2*cosh(im(x))*sin(re(x)) + re(c))*sin(re(x))*sinh(im(x))\   (1 + cos(re(x))*cosh(im(x)))*(2*cosh(im(x))*sin(re(x)) + re(c))   (2*cos(re(x))*sinh(im(x)) + im(c))*sin(re(x))*sinh(im(x))
y1 = I*|--------------------------------------------------------------- + ---------------------------------------------------------| + --------------------------------------------------------------- - ---------------------------------------------------------
       |                                2      2            2                                          2      2            2       |                                   2      2            2                                          2      2            2       
       \    (1 + cos(re(x))*cosh(im(x)))  + sin (re(x))*sinh (im(x))       (1 + cos(re(x))*cosh(im(x)))  + sin (re(x))*sinh (im(x))/       (1 + cos(re(x))*cosh(im(x)))  + sin (re(x))*sinh (im(x))       (1 + cos(re(x))*cosh(im(x)))  + sin (re(x))*sinh (im(x))
$$y_{1} = \frac{\left(2 \sin{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} + \operatorname{re}{\left(c\right)}\right) \left(\cos{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} + 1\right)}{\left(\cos{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} + 1\right)^{2} + \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)}} - \frac{\left(2 \cos{\left(\operatorname{re}{\left(x\right)} \right)} \sinh{\left(\operatorname{im}{\left(x\right)} \right)} + \operatorname{im}{\left(c\right)}\right) \sin{\left(\operatorname{re}{\left(x\right)} \right)} \sinh{\left(\operatorname{im}{\left(x\right)} \right)}}{\left(\cos{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} + 1\right)^{2} + \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)}} + i \left(\frac{\left(2 \sin{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} + \operatorname{re}{\left(c\right)}\right) \sin{\left(\operatorname{re}{\left(x\right)} \right)} \sinh{\left(\operatorname{im}{\left(x\right)} \right)}}{\left(\cos{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} + 1\right)^{2} + \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)}} + \frac{\left(2 \cos{\left(\operatorname{re}{\left(x\right)} \right)} \sinh{\left(\operatorname{im}{\left(x\right)} \right)} + \operatorname{im}{\left(c\right)}\right) \left(\cos{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} + 1\right)}{\left(\cos{\left(\operatorname{re}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)} + 1\right)^{2} + \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)}}\right)$$ 
y1 = (2*sin(re(x))*cosh(im(x)) + re(c))*(cos(re(x))*cosh(im(x)) + 1)/((cos(re(x))*cosh(im(x)) + 1)^2 + sin(re(x))^2*sinh(im(x))^2) - (2*cos(re(x))*sinh(im(x)) + im(c))*sin(re(x))*sinh(im(x))/((cos(re(x))*cosh(im(x)) + 1)^2 + sin(re(x))^2*sinh(im(x))^2) + i*((2*sin(re(x))*cosh(im(x)) + re(c))*sin(re(x))*sinh(im(x))/((cos(re(x))*cosh(im(x)) + 1)^2 + sin(re(x))^2*sinh(im(x))^2) + (2*cos(re(x))*sinh(im(x)) + im(c))*(cos(re(x))*cosh(im(x)) + 1)/((cos(re(x))*cosh(im(x)) + 1)^2 + sin(re(x))^2*sinh(im(x))^2))
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