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-y-log(y)=Const-((x^2)/2)+x la ecuación

El profesor se sorprenderá mucho al ver tu solución correcta😉

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Solución numérica:

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Solución

Ha introducido [src] 
                   2    
                  x     
-y - log(y) = c - -- + x
                  2
−y−log⁡(y)=x+(c−x22)- y - \log{\left(y \right)} = x + \left(c - \frac{x^{2}}{2}\right)
Simplificar
Solución detallada
Transportemos el miembro derecho de la ecuación al
miembro izquierdo de la ecuación con el signo negativo.

La ecuación se convierte de
−y−log⁡(y)=x+(c−x22)- y - \log{\left(y \right)} = x + \left(c - \frac{x^{2}}{2}\right)
en
(−x+(−c+x22))+(−y−log⁡(y))=0\left(- x + \left(- c + \frac{x^{2}}{2}\right)\right) + \left(- y - \log{\left(y \right)}\right) = 0
Abramos la expresión en la ecuación
(−x+(−c+x22))+(−y−log⁡(y))=0\left(- x + \left(- c + \frac{x^{2}}{2}\right)\right) + \left(- y - \log{\left(y \right)}\right) = 0
Obtenemos la ecuación cuadrática
−c+x22−x−y−log⁡(y)=0- c + \frac{x^{2}}{2} - x - y - \log{\left(y \right)} = 0
Es la ecuación de la forma
a*x^2 + b*x + c = 0

La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
x1=D−b2ax_{1} = \frac{\sqrt{D} - b}{2 a}
x2=−D−b2ax_{2} = \frac{- \sqrt{D} - b}{2 a}
donde D = b^2 - 4*a*c es el discriminante.
Como
a=12a = \frac{1}{2}
b=−1b = -1
c=−c−y−log⁡(y)c = - c - y - \log{\left(y \right)}
, entonces
D = b^2 - 4 * a * c = 

(-1)^2 - 4 * (1/2) * (-c - y - log(y)) = 1 + 2*c + 2*y + 2*log(y)

La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

o
x1=2c+2y+2log⁡(y)+1+1x_{1} = \sqrt{2 c + 2 y + 2 \log{\left(y \right)} + 1} + 1
x2=1−2c+2y+2log⁡(y)+1x_{2} = 1 - \sqrt{2 c + 2 y + 2 \log{\left(y \right)} + 1}
Teorema de Cardano-Vieta
reescribamos la ecuación
−y−log⁡(y)=x+(c−x22)- y - \log{\left(y \right)} = x + \left(c - \frac{x^{2}}{2}\right)
de
ax2+bx+c=0a x^{2} + b x + c = 0
como ecuación cuadrática reducida
x2+bxa+ca=0x^{2} + \frac{b x}{a} + \frac{c}{a} = 0
−2c+x2−2x−2y−2log⁡(y)=0- 2 c + x^{2} - 2 x - 2 y - 2 \log{\left(y \right)} = 0
px+q+x2=0p x + q + x^{2} = 0
donde
p=bap = \frac{b}{a}
p=−2p = -2
q=caq = \frac{c}{a}
q=−2c−2y−2log⁡(y)q = - 2 c - 2 y - 2 \log{\left(y \right)}
Fórmulas de Cardano-Vieta
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=2x_{1} + x_{2} = 2
x1x2=−2c−2y−2log⁡(y)x_{1} x_{2} = - 2 c - 2 y - 2 \log{\left(y \right)}
Gráfica
Respuesta rápida [src] 
            _________________________________________________________________________                                                                                     _________________________________________________________________________                                                                             
         4 /                               2                                       2     /atan2(2*arg(y) + 2*im(c) + 2*im(y), 1 + 2*log(|y|) + 2*re(c) + 2*re(y))\     4 /                               2                                       2     /atan2(2*arg(y) + 2*im(c) + 2*im(y), 1 + 2*log(|y|) + 2*re(c) + 2*re(y))\
x1 = 1 - \/  (2*arg(y) + 2*im(c) + 2*im(y))  + (1 + 2*log(|y|) + 2*re(c) + 2*re(y))  *cos|-----------------------------------------------------------------------| - I*\/  (2*arg(y) + 2*im(c) + 2*im(y))  + (1 + 2*log(|y|) + 2*re(c) + 2*re(y))  *sin|-----------------------------------------------------------------------|
                                                                                         \                                   2                                   /                                                                                     \                                   2                                   /
x1=−i(2im⁡(c)+2im⁡(y)+2arg⁡(y))2+(2log⁡(∣y∣)+2re⁡(c)+2re⁡(y)+1)24sin⁡(atan2⁡(2im⁡(c)+2im⁡(y)+2arg⁡(y),2log⁡(∣y∣)+2re⁡(c)+2re⁡(y)+1)2)−(2im⁡(c)+2im⁡(y)+2arg⁡(y))2+(2log⁡(∣y∣)+2re⁡(c)+2re⁡(y)+1)24cos⁡(atan2⁡(2im⁡(c)+2im⁡(y)+2arg⁡(y),2log⁡(∣y∣)+2re⁡(c)+2re⁡(y)+1)2)+1x_{1} = - i \sqrt[4]{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)}\right)^{2} + \left(2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1 \right)}}{2} \right)} - \sqrt[4]{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)}\right)^{2} + \left(2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1 \right)}}{2} \right)} + 1 
            _________________________________________________________________________                                                                                     _________________________________________________________________________                                                                             
         4 /                               2                                       2     /atan2(2*arg(y) + 2*im(c) + 2*im(y), 1 + 2*log(|y|) + 2*re(c) + 2*re(y))\     4 /                               2                                       2     /atan2(2*arg(y) + 2*im(c) + 2*im(y), 1 + 2*log(|y|) + 2*re(c) + 2*re(y))\
x2 = 1 + \/  (2*arg(y) + 2*im(c) + 2*im(y))  + (1 + 2*log(|y|) + 2*re(c) + 2*re(y))  *cos|-----------------------------------------------------------------------| + I*\/  (2*arg(y) + 2*im(c) + 2*im(y))  + (1 + 2*log(|y|) + 2*re(c) + 2*re(y))  *sin|-----------------------------------------------------------------------|
                                                                                         \                                   2                                   /                                                                                     \                                   2                                   /
x2=i(2im⁡(c)+2im⁡(y)+2arg⁡(y))2+(2log⁡(∣y∣)+2re⁡(c)+2re⁡(y)+1)24sin⁡(atan2⁡(2im⁡(c)+2im⁡(y)+2arg⁡(y),2log⁡(∣y∣)+2re⁡(c)+2re⁡(y)+1)2)+(2im⁡(c)+2im⁡(y)+2arg⁡(y))2+(2log⁡(∣y∣)+2re⁡(c)+2re⁡(y)+1)24cos⁡(atan2⁡(2im⁡(c)+2im⁡(y)+2arg⁡(y),2log⁡(∣y∣)+2re⁡(c)+2re⁡(y)+1)2)+1x_{2} = i \sqrt[4]{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)}\right)^{2} + \left(2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1 \right)}}{2} \right)} + \sqrt[4]{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)}\right)^{2} + \left(2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1 \right)}}{2} \right)} + 1 
x2 = i*((2*im(c) + 2*im(y) + 2*arg(y))^2 + (2*log(|y|) + 2*re(c) + 2*re(y) + 1)^2)^(1/4)*sin(atan2(2*im(c) + 2*im(y) + 2*arg(y, 2*log(|y|) + 2*re(c) + 2*re(y) + 1)/2) + ((2*im(c) + 2*im(y) + 2*arg(y))^2 + (2*log(|y|) + 2*re(c) + 2*re(y) + 1)^2)^(1/4)*cos(atan2(2*im(c) + 2*im(y) + 2*arg(y), 2*log(|y|) + 2*re(c) + 2*re(y) + 1)/2) + 1)
Suma y producto de raíces [src] 
suma
       _________________________________________________________________________                                                                                     _________________________________________________________________________                                                                                       _________________________________________________________________________                                                                                     _________________________________________________________________________                                                                             
    4 /                               2                                       2     /atan2(2*arg(y) + 2*im(c) + 2*im(y), 1 + 2*log(|y|) + 2*re(c) + 2*re(y))\     4 /                               2                                       2     /atan2(2*arg(y) + 2*im(c) + 2*im(y), 1 + 2*log(|y|) + 2*re(c) + 2*re(y))\       4 /                               2                                       2     /atan2(2*arg(y) + 2*im(c) + 2*im(y), 1 + 2*log(|y|) + 2*re(c) + 2*re(y))\     4 /                               2                                       2     /atan2(2*arg(y) + 2*im(c) + 2*im(y), 1 + 2*log(|y|) + 2*re(c) + 2*re(y))\
1 - \/  (2*arg(y) + 2*im(c) + 2*im(y))  + (1 + 2*log(|y|) + 2*re(c) + 2*re(y))  *cos|-----------------------------------------------------------------------| - I*\/  (2*arg(y) + 2*im(c) + 2*im(y))  + (1 + 2*log(|y|) + 2*re(c) + 2*re(y))  *sin|-----------------------------------------------------------------------| + 1 + \/  (2*arg(y) + 2*im(c) + 2*im(y))  + (1 + 2*log(|y|) + 2*re(c) + 2*re(y))  *cos|-----------------------------------------------------------------------| + I*\/  (2*arg(y) + 2*im(c) + 2*im(y))  + (1 + 2*log(|y|) + 2*re(c) + 2*re(y))  *sin|-----------------------------------------------------------------------|
                                                                                    \                                   2                                   /                                                                                     \                                   2                                   /                                                                                       \                                   2                                   /                                                                                     \                                   2                                   /
(−i(2im⁡(c)+2im⁡(y)+2arg⁡(y))2+(2log⁡(∣y∣)+2re⁡(c)+2re⁡(y)+1)24sin⁡(atan2⁡(2im⁡(c)+2im⁡(y)+2arg⁡(y),2log⁡(∣y∣)+2re⁡(c)+2re⁡(y)+1)2)−(2im⁡(c)+2im⁡(y)+2arg⁡(y))2+(2log⁡(∣y∣)+2re⁡(c)+2re⁡(y)+1)24cos⁡(atan2⁡(2im⁡(c)+2im⁡(y)+2arg⁡(y),2log⁡(∣y∣)+2re⁡(c)+2re⁡(y)+1)2)+1)+(i(2im⁡(c)+2im⁡(y)+2arg⁡(y))2+(2log⁡(∣y∣)+2re⁡(c)+2re⁡(y)+1)24sin⁡(atan2⁡(2im⁡(c)+2im⁡(y)+2arg⁡(y),2log⁡(∣y∣)+2re⁡(c)+2re⁡(y)+1)2)+(2im⁡(c)+2im⁡(y)+2arg⁡(y))2+(2log⁡(∣y∣)+2re⁡(c)+2re⁡(y)+1)24cos⁡(atan2⁡(2im⁡(c)+2im⁡(y)+2arg⁡(y),2log⁡(∣y∣)+2re⁡(c)+2re⁡(y)+1)2)+1)\left(- i \sqrt[4]{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)}\right)^{2} + \left(2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1 \right)}}{2} \right)} - \sqrt[4]{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)}\right)^{2} + \left(2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1 \right)}}{2} \right)} + 1\right) + \left(i \sqrt[4]{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)}\right)^{2} + \left(2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1 \right)}}{2} \right)} + \sqrt[4]{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)}\right)^{2} + \left(2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1 \right)}}{2} \right)} + 1\right) 
=
2
22 
producto
/       _________________________________________________________________________                                                                                     _________________________________________________________________________                                                                             \ /       _________________________________________________________________________                                                                                     _________________________________________________________________________                                                                             \
|    4 /                               2                                       2     /atan2(2*arg(y) + 2*im(c) + 2*im(y), 1 + 2*log(|y|) + 2*re(c) + 2*re(y))\     4 /                               2                                       2     /atan2(2*arg(y) + 2*im(c) + 2*im(y), 1 + 2*log(|y|) + 2*re(c) + 2*re(y))\| |    4 /                               2                                       2     /atan2(2*arg(y) + 2*im(c) + 2*im(y), 1 + 2*log(|y|) + 2*re(c) + 2*re(y))\     4 /                               2                                       2     /atan2(2*arg(y) + 2*im(c) + 2*im(y), 1 + 2*log(|y|) + 2*re(c) + 2*re(y))\|
|1 - \/  (2*arg(y) + 2*im(c) + 2*im(y))  + (1 + 2*log(|y|) + 2*re(c) + 2*re(y))  *cos|-----------------------------------------------------------------------| - I*\/  (2*arg(y) + 2*im(c) + 2*im(y))  + (1 + 2*log(|y|) + 2*re(c) + 2*re(y))  *sin|-----------------------------------------------------------------------||*|1 + \/  (2*arg(y) + 2*im(c) + 2*im(y))  + (1 + 2*log(|y|) + 2*re(c) + 2*re(y))  *cos|-----------------------------------------------------------------------| + I*\/  (2*arg(y) + 2*im(c) + 2*im(y))  + (1 + 2*log(|y|) + 2*re(c) + 2*re(y))  *sin|-----------------------------------------------------------------------||
\                                                                                    \                                   2                                   /                                                                                     \                                   2                                   // \                                                                                    \                                   2                                   /                                                                                     \                                   2                                   //
(−i(2im⁡(c)+2im⁡(y)+2arg⁡(y))2+(2log⁡(∣y∣)+2re⁡(c)+2re⁡(y)+1)24sin⁡(atan2⁡(2im⁡(c)+2im⁡(y)+2arg⁡(y),2log⁡(∣y∣)+2re⁡(c)+2re⁡(y)+1)2)−(2im⁡(c)+2im⁡(y)+2arg⁡(y))2+(2log⁡(∣y∣)+2re⁡(c)+2re⁡(y)+1)24cos⁡(atan2⁡(2im⁡(c)+2im⁡(y)+2arg⁡(y),2log⁡(∣y∣)+2re⁡(c)+2re⁡(y)+1)2)+1)(i(2im⁡(c)+2im⁡(y)+2arg⁡(y))2+(2log⁡(∣y∣)+2re⁡(c)+2re⁡(y)+1)24sin⁡(atan2⁡(2im⁡(c)+2im⁡(y)+2arg⁡(y),2log⁡(∣y∣)+2re⁡(c)+2re⁡(y)+1)2)+(2im⁡(c)+2im⁡(y)+2arg⁡(y))2+(2log⁡(∣y∣)+2re⁡(c)+2re⁡(y)+1)24cos⁡(atan2⁡(2im⁡(c)+2im⁡(y)+2arg⁡(y),2log⁡(∣y∣)+2re⁡(c)+2re⁡(y)+1)2)+1)\left(- i \sqrt[4]{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)}\right)^{2} + \left(2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1 \right)}}{2} \right)} - \sqrt[4]{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)}\right)^{2} + \left(2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1 \right)}}{2} \right)} + 1\right) \left(i \sqrt[4]{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)}\right)^{2} + \left(2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1 \right)}}{2} \right)} + \sqrt[4]{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)}\right)^{2} + \left(2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1 \right)}}{2} \right)} + 1\right) 
=
-2*log(|y|) - 2*re(c) - 2*re(y) - 2*I*arg(y) - 2*I*im(c) - 2*I*im(y)
−2log⁡(∣y∣)−2re⁡(c)−2re⁡(y)−2iim⁡(c)−2iim⁡(y)−2iarg⁡(y)- 2 \log{\left(\left|{y}\right| \right)} - 2 \operatorname{re}{\left(c\right)} - 2 \operatorname{re}{\left(y\right)} - 2 i \operatorname{im}{\left(c\right)} - 2 i \operatorname{im}{\left(y\right)} - 2 i \arg{\left(y \right)} 
-2*log(|y|) - 2*re(c) - 2*re(y) - 2*i*arg(y) - 2*i*im(c) - 2*i*im(y)
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