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ln|y|-y=ln|x|+x/2+C 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] 
                          x    
log(|y|) - y = log(|x|) + - + c
                          2
−y+log⁡(∣y∣)=c+(x2+log⁡(∣x∣))- y + \log{\left(\left|{y}\right| \right)} = c + \left(\frac{x}{2} + \log{\left(\left|{x}\right| \right)}\right)
Simplificar
Gráfica
Respuesta rápida [src] 
         //  /    __________    \       /    __________    \     \     //  /    __________    \       /    __________    \     \
         ||  |   /  x + 2*c     |       |   /  x + 2*c     |     |     ||  |   /  x + 2*c     |       |   /  x + 2*c     |     |
y1 = I*im|<-W\-\/  e        *|x|/  for W\-\/  e        *|x|/ <= 0| + re|<-W\-\/  e        *|x|/  for W\-\/  e        *|x|/ <= 0|
         ||                                                      |     ||                                                      |
         \\         nan                      otherwise           /     \\         nan                      otherwise           /
y1=re⁡({−W(−e2c+x∣x∣)for W(−e2c+x∣x∣)≤0NaNotherwise)+iim⁡({−W(−e2c+x∣x∣)for W(−e2c+x∣x∣)≤0NaNotherwise)y_{1} = \operatorname{re}{\left(\begin{cases} - W\left(- \sqrt{e^{2 c + x}} \left|{x}\right|\right) & \text{for}\: W\left(- \sqrt{e^{2 c + x}} \left|{x}\right|\right) \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - W\left(- \sqrt{e^{2 c + x}} \left|{x}\right|\right) & \text{for}\: W\left(- \sqrt{e^{2 c + x}} \left|{x}\right|\right) \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} 
         //  /    __________    \       /    __________    \    \     //  /    __________    \       /    __________    \    \
         ||  |   /  x + 2*c     |       |   /  x + 2*c     |    |     ||  |   /  x + 2*c     |       |   /  x + 2*c     |    |
y2 = I*im|<-W\-\/  e        *|x|/  for W\-\/  e        *|x|/ > 0| + re|<-W\-\/  e        *|x|/  for W\-\/  e        *|x|/ > 0|
         ||                                                     |     ||                                                     |
         \\         nan                      otherwise          /     \\         nan                      otherwise          /
y2=re⁡({−W(−e2c+x∣x∣)for W(−e2c+x∣x∣)>0NaNotherwise)+iim⁡({−W(−e2c+x∣x∣)for W(−e2c+x∣x∣)>0NaNotherwise)y_{2} = \operatorname{re}{\left(\begin{cases} - W\left(- \sqrt{e^{2 c + x}} \left|{x}\right|\right) & \text{for}\: W\left(- \sqrt{e^{2 c + x}} \left|{x}\right|\right) > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - W\left(- \sqrt{e^{2 c + x}} \left|{x}\right|\right) & \text{for}\: W\left(- \sqrt{e^{2 c + x}} \left|{x}\right|\right) > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} 
         //  /   __________    \       /   __________    \     \     //  /   __________    \       /   __________    \     \
         ||  |  /  x + 2*c     |       |  /  x + 2*c     |     |     ||  |  /  x + 2*c     |       |  /  x + 2*c     |     |
y3 = I*im|<-W\\/  e        *|x|/  for W\\/  e        *|x|/ <= 0| + re|<-W\\/  e        *|x|/  for W\\/  e        *|x|/ <= 0|
         ||                                                    |     ||                                                    |
         \\         nan                     otherwise          /     \\         nan                     otherwise          /
y3=re⁡({−W(e2c+x∣x∣)for W(e2c+x∣x∣)≤0NaNotherwise)+iim⁡({−W(e2c+x∣x∣)for W(e2c+x∣x∣)≤0NaNotherwise)y_{3} = \operatorname{re}{\left(\begin{cases} - W\left(\sqrt{e^{2 c + x}} \left|{x}\right|\right) & \text{for}\: W\left(\sqrt{e^{2 c + x}} \left|{x}\right|\right) \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - W\left(\sqrt{e^{2 c + x}} \left|{x}\right|\right) & \text{for}\: W\left(\sqrt{e^{2 c + x}} \left|{x}\right|\right) \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} 
         //  /   __________    \       /   __________    \    \     //  /   __________    \       /   __________    \    \
         ||  |  /  x + 2*c     |       |  /  x + 2*c     |    |     ||  |  /  x + 2*c     |       |  /  x + 2*c     |    |
y4 = I*im|<-W\\/  e        *|x|/  for W\\/  e        *|x|/ > 0| + re|<-W\\/  e        *|x|/  for W\\/  e        *|x|/ > 0|
         ||                                                   |     ||                                                   |
         \\         nan                    otherwise          /     \\         nan                    otherwise          /
y4=re⁡({−W(e2c+x∣x∣)for W(e2c+x∣x∣)>0NaNotherwise)+iim⁡({−W(e2c+x∣x∣)for W(e2c+x∣x∣)>0NaNotherwise)y_{4} = \operatorname{re}{\left(\begin{cases} - W\left(\sqrt{e^{2 c + x}} \left|{x}\right|\right) & \text{for}\: W\left(\sqrt{e^{2 c + x}} \left|{x}\right|\right) > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - W\left(\sqrt{e^{2 c + x}} \left|{x}\right|\right) & \text{for}\: W\left(\sqrt{e^{2 c + x}} \left|{x}\right|\right) > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} 
y4 = re(Piecewise((-LambertW(sqrt(exp(2*c + x))*|x|, LambertW(sqrt(exp(2*c + x))*|x|) > 0), (nan, True))) + i*im(Piecewise((-LambertW(sqrt(exp(2*c + x))*|x|), LambertW(sqrt(exp(2*c + x))*|x|) > 0), (nan, True))))
Suma y producto de raíces [src] 
suma
    //  /    __________    \       /    __________    \     \     //  /    __________    \       /    __________    \     \       //  /    __________    \       /    __________    \    \     //  /    __________    \       /    __________    \    \       //  /   __________    \       /   __________    \     \     //  /   __________    \       /   __________    \     \       //  /   __________    \       /   __________    \    \     //  /   __________    \       /   __________    \    \
    ||  |   /  x + 2*c     |       |   /  x + 2*c     |     |     ||  |   /  x + 2*c     |       |   /  x + 2*c     |     |       ||  |   /  x + 2*c     |       |   /  x + 2*c     |    |     ||  |   /  x + 2*c     |       |   /  x + 2*c     |    |       ||  |  /  x + 2*c     |       |  /  x + 2*c     |     |     ||  |  /  x + 2*c     |       |  /  x + 2*c     |     |       ||  |  /  x + 2*c     |       |  /  x + 2*c     |    |     ||  |  /  x + 2*c     |       |  /  x + 2*c     |    |
I*im|<-W\-\/  e        *|x|/  for W\-\/  e        *|x|/ <= 0| + re|<-W\-\/  e        *|x|/  for W\-\/  e        *|x|/ <= 0| + I*im|<-W\-\/  e        *|x|/  for W\-\/  e        *|x|/ > 0| + re|<-W\-\/  e        *|x|/  for W\-\/  e        *|x|/ > 0| + I*im|<-W\\/  e        *|x|/  for W\\/  e        *|x|/ <= 0| + re|<-W\\/  e        *|x|/  for W\\/  e        *|x|/ <= 0| + I*im|<-W\\/  e        *|x|/  for W\\/  e        *|x|/ > 0| + re|<-W\\/  e        *|x|/  for W\\/  e        *|x|/ > 0|
    ||                                                      |     ||                                                      |       ||                                                     |     ||                                                     |       ||                                                    |     ||                                                    |       ||                                                   |     ||                                                   |
    \\         nan                      otherwise           /     \\         nan                      otherwise           /       \\         nan                      otherwise          /     \\         nan                      otherwise          /       \\         nan                     otherwise          /     \\         nan                     otherwise          /       \\         nan                    otherwise          /     \\         nan                    otherwise          /
(((re⁡({−W(−e2c+x∣x∣)for W(−e2c+x∣x∣)≤0NaNotherwise)+iim⁡({−W(−e2c+x∣x∣)for W(−e2c+x∣x∣)≤0NaNotherwise))+(re⁡({−W(−e2c+x∣x∣)for W(−e2c+x∣x∣)>0NaNotherwise)+iim⁡({−W(−e2c+x∣x∣)for W(−e2c+x∣x∣)>0NaNotherwise)))+(re⁡({−W(e2c+x∣x∣)for W(e2c+x∣x∣)≤0NaNotherwise)+iim⁡({−W(e2c+x∣x∣)for W(e2c+x∣x∣)≤0NaNotherwise)))+(re⁡({−W(e2c+x∣x∣)for W(e2c+x∣x∣)>0NaNotherwise)+iim⁡({−W(e2c+x∣x∣)for W(e2c+x∣x∣)>0NaNotherwise))\left(\left(\left(\operatorname{re}{\left(\begin{cases} - W\left(- \sqrt{e^{2 c + x}} \left|{x}\right|\right) & \text{for}\: W\left(- \sqrt{e^{2 c + x}} \left|{x}\right|\right) \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - W\left(- \sqrt{e^{2 c + x}} \left|{x}\right|\right) & \text{for}\: W\left(- \sqrt{e^{2 c + x}} \left|{x}\right|\right) \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) + \left(\operatorname{re}{\left(\begin{cases} - W\left(- \sqrt{e^{2 c + x}} \left|{x}\right|\right) & \text{for}\: W\left(- \sqrt{e^{2 c + x}} \left|{x}\right|\right) > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - W\left(- \sqrt{e^{2 c + x}} \left|{x}\right|\right) & \text{for}\: W\left(- \sqrt{e^{2 c + x}} \left|{x}\right|\right) > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right)\right) + \left(\operatorname{re}{\left(\begin{cases} - W\left(\sqrt{e^{2 c + x}} \left|{x}\right|\right) & \text{for}\: W\left(\sqrt{e^{2 c + x}} \left|{x}\right|\right) \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - W\left(\sqrt{e^{2 c + x}} \left|{x}\right|\right) & \text{for}\: W\left(\sqrt{e^{2 c + x}} \left|{x}\right|\right) \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right)\right) + \left(\operatorname{re}{\left(\begin{cases} - W\left(\sqrt{e^{2 c + x}} \left|{x}\right|\right) & \text{for}\: W\left(\sqrt{e^{2 c + x}} \left|{x}\right|\right) > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - W\left(\sqrt{e^{2 c + x}} \left|{x}\right|\right) & \text{for}\: W\left(\sqrt{e^{2 c + x}} \left|{x}\right|\right) > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) 
=
    //  /   __________    \       /   __________    \    \       //  /   __________    \       /   __________    \     \       //  /    __________    \       /    __________    \    \       //  /    __________    \       /    __________    \     \     //  /   __________    \       /   __________    \    \     //  /   __________    \       /   __________    \     \     //  /    __________    \       /    __________    \    \     //  /    __________    \       /    __________    \     \
    ||  |  /  x + 2*c     |       |  /  x + 2*c     |    |       ||  |  /  x + 2*c     |       |  /  x + 2*c     |     |       ||  |   /  x + 2*c     |       |   /  x + 2*c     |    |       ||  |   /  x + 2*c     |       |   /  x + 2*c     |     |     ||  |  /  x + 2*c     |       |  /  x + 2*c     |    |     ||  |  /  x + 2*c     |       |  /  x + 2*c     |     |     ||  |   /  x + 2*c     |       |   /  x + 2*c     |    |     ||  |   /  x + 2*c     |       |   /  x + 2*c     |     |
I*im|<-W\\/  e        *|x|/  for W\\/  e        *|x|/ > 0| + I*im|<-W\\/  e        *|x|/  for W\\/  e        *|x|/ <= 0| + I*im|<-W\-\/  e        *|x|/  for W\-\/  e        *|x|/ > 0| + I*im|<-W\-\/  e        *|x|/  for W\-\/  e        *|x|/ <= 0| + re|<-W\\/  e        *|x|/  for W\\/  e        *|x|/ > 0| + re|<-W\\/  e        *|x|/  for W\\/  e        *|x|/ <= 0| + re|<-W\-\/  e        *|x|/  for W\-\/  e        *|x|/ > 0| + re|<-W\-\/  e        *|x|/  for W\-\/  e        *|x|/ <= 0|
    ||                                                   |       ||                                                    |       ||                                                     |       ||                                                      |     ||                                                   |     ||                                                    |     ||                                                     |     ||                                                      |
    \\         nan                    otherwise          /       \\         nan                     otherwise          /       \\         nan                      otherwise          /       \\         nan                      otherwise           /     \\         nan                    otherwise          /     \\         nan                     otherwise          /     \\         nan                      otherwise          /     \\         nan                      otherwise           /
re⁡({−W(−e2c+x∣x∣)for W(−e2c+x∣x∣)≤0NaNotherwise)+re⁡({−W(−e2c+x∣x∣)for W(−e2c+x∣x∣)>0NaNotherwise)+re⁡({−W(e2c+x∣x∣)for W(e2c+x∣x∣)≤0NaNotherwise)+re⁡({−W(e2c+x∣x∣)for W(e2c+x∣x∣)>0NaNotherwise)+iim⁡({−W(−e2c+x∣x∣)for W(−e2c+x∣x∣)≤0NaNotherwise)+iim⁡({−W(−e2c+x∣x∣)for W(−e2c+x∣x∣)>0NaNotherwise)+iim⁡({−W(e2c+x∣x∣)for W(e2c+x∣x∣)≤0NaNotherwise)+iim⁡({−W(e2c+x∣x∣)for W(e2c+x∣x∣)>0NaNotherwise)\operatorname{re}{\left(\begin{cases} - W\left(- \sqrt{e^{2 c + x}} \left|{x}\right|\right) & \text{for}\: W\left(- \sqrt{e^{2 c + x}} \left|{x}\right|\right) \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + \operatorname{re}{\left(\begin{cases} - W\left(- \sqrt{e^{2 c + x}} \left|{x}\right|\right) & \text{for}\: W\left(- \sqrt{e^{2 c + x}} \left|{x}\right|\right) > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + \operatorname{re}{\left(\begin{cases} - W\left(\sqrt{e^{2 c + x}} \left|{x}\right|\right) & \text{for}\: W\left(\sqrt{e^{2 c + x}} \left|{x}\right|\right) \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + \operatorname{re}{\left(\begin{cases} - W\left(\sqrt{e^{2 c + x}} \left|{x}\right|\right) & \text{for}\: W\left(\sqrt{e^{2 c + x}} \left|{x}\right|\right) > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - W\left(- \sqrt{e^{2 c + x}} \left|{x}\right|\right) & \text{for}\: W\left(- \sqrt{e^{2 c + x}} \left|{x}\right|\right) \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - W\left(- \sqrt{e^{2 c + x}} \left|{x}\right|\right) & \text{for}\: W\left(- \sqrt{e^{2 c + x}} \left|{x}\right|\right) > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - W\left(\sqrt{e^{2 c + x}} \left|{x}\right|\right) & \text{for}\: W\left(\sqrt{e^{2 c + x}} \left|{x}\right|\right) \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - W\left(\sqrt{e^{2 c + x}} \left|{x}\right|\right) & \text{for}\: W\left(\sqrt{e^{2 c + x}} \left|{x}\right|\right) > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} 
producto
/    //  /    __________    \       /    __________    \     \     //  /    __________    \       /    __________    \     \\ /    //  /    __________    \       /    __________    \    \     //  /    __________    \       /    __________    \    \\ /    //  /   __________    \       /   __________    \     \     //  /   __________    \       /   __________    \     \\ /    //  /   __________    \       /   __________    \    \     //  /   __________    \       /   __________    \    \\
|    ||  |   /  x + 2*c     |       |   /  x + 2*c     |     |     ||  |   /  x + 2*c     |       |   /  x + 2*c     |     || |    ||  |   /  x + 2*c     |       |   /  x + 2*c     |    |     ||  |   /  x + 2*c     |       |   /  x + 2*c     |    || |    ||  |  /  x + 2*c     |       |  /  x + 2*c     |     |     ||  |  /  x + 2*c     |       |  /  x + 2*c     |     || |    ||  |  /  x + 2*c     |       |  /  x + 2*c     |    |     ||  |  /  x + 2*c     |       |  /  x + 2*c     |    ||
|I*im|<-W\-\/  e        *|x|/  for W\-\/  e        *|x|/ <= 0| + re|<-W\-\/  e        *|x|/  for W\-\/  e        *|x|/ <= 0||*|I*im|<-W\-\/  e        *|x|/  for W\-\/  e        *|x|/ > 0| + re|<-W\-\/  e        *|x|/  for W\-\/  e        *|x|/ > 0||*|I*im|<-W\\/  e        *|x|/  for W\\/  e        *|x|/ <= 0| + re|<-W\\/  e        *|x|/  for W\\/  e        *|x|/ <= 0||*|I*im|<-W\\/  e        *|x|/  for W\\/  e        *|x|/ > 0| + re|<-W\\/  e        *|x|/  for W\\/  e        *|x|/ > 0||
|    ||                                                      |     ||                                                      || |    ||                                                     |     ||                                                     || |    ||                                                    |     ||                                                    || |    ||                                                   |     ||                                                   ||
\    \\         nan                      otherwise           /     \\         nan                      otherwise           // \    \\         nan                      otherwise          /     \\         nan                      otherwise          // \    \\         nan                     otherwise          /     \\         nan                     otherwise          // \    \\         nan                    otherwise          /     \\         nan                    otherwise          //
(re⁡({−W(−e2c+x∣x∣)for W(−e2c+x∣x∣)≤0NaNotherwise)+iim⁡({−W(−e2c+x∣x∣)for W(−e2c+x∣x∣)≤0NaNotherwise))(re⁡({−W(−e2c+x∣x∣)for W(−e2c+x∣x∣)>0NaNotherwise)+iim⁡({−W(−e2c+x∣x∣)for W(−e2c+x∣x∣)>0NaNotherwise))(re⁡({−W(e2c+x∣x∣)for W(e2c+x∣x∣)≤0NaNotherwise)+iim⁡({−W(e2c+x∣x∣)for W(e2c+x∣x∣)≤0NaNotherwise))(re⁡({−W(e2c+x∣x∣)for W(e2c+x∣x∣)>0NaNotherwise)+iim⁡({−W(e2c+x∣x∣)for W(e2c+x∣x∣)>0NaNotherwise))\left(\operatorname{re}{\left(\begin{cases} - W\left(- \sqrt{e^{2 c + x}} \left|{x}\right|\right) & \text{for}\: W\left(- \sqrt{e^{2 c + x}} \left|{x}\right|\right) \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - W\left(- \sqrt{e^{2 c + x}} \left|{x}\right|\right) & \text{for}\: W\left(- \sqrt{e^{2 c + x}} \left|{x}\right|\right) \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) \left(\operatorname{re}{\left(\begin{cases} - W\left(- \sqrt{e^{2 c + x}} \left|{x}\right|\right) & \text{for}\: W\left(- \sqrt{e^{2 c + x}} \left|{x}\right|\right) > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - W\left(- \sqrt{e^{2 c + x}} \left|{x}\right|\right) & \text{for}\: W\left(- \sqrt{e^{2 c + x}} \left|{x}\right|\right) > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) \left(\operatorname{re}{\left(\begin{cases} - W\left(\sqrt{e^{2 c + x}} \left|{x}\right|\right) & \text{for}\: W\left(\sqrt{e^{2 c + x}} \left|{x}\right|\right) \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - W\left(\sqrt{e^{2 c + x}} \left|{x}\right|\right) & \text{for}\: W\left(\sqrt{e^{2 c + x}} \left|{x}\right|\right) \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) \left(\operatorname{re}{\left(\begin{cases} - W\left(\sqrt{e^{2 c + x}} \left|{x}\right|\right) & \text{for}\: W\left(\sqrt{e^{2 c + x}} \left|{x}\right|\right) > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - W\left(\sqrt{e^{2 c + x}} \left|{x}\right|\right) & \text{for}\: W\left(\sqrt{e^{2 c + x}} \left|{x}\right|\right) > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) 
=
nan
NaN\text{NaN} 
nan
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