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lny=-2ln(x^2+1)+const la ecuación

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Solución numérica:

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Solución

Ha introducido [src]
                / 2    \    
log(y) = - 2*log\x  + 1/ + c
$$\log{\left(y \right)} = c - 2 \log{\left(x^{2} + 1 \right)}$$
Gráfica
Respuesta rápida [src]
                                                    /     /  /   c \         /   c \\\                                                  /     /  /   c \         /   c \\\
                ________________________________    |     |  |   - |         |   - |||              ________________________________    |     |  |   - |         |   - |||
               /                 2                  |     |  |   2 |         |   2 |||             /                 2                  |     |  |   2 |         |   2 |||
              /  /       /   c \\       /   c \     |     |  |  e  |         |  e  |||            /  /       /   c \\       /   c \     |     |  |  e  |         |  e  |||
             /   |       |   - ||       |   - |     |atan2|im|-----|, -1 + re|-----|||           /   |       |   - ||       |   - |     |atan2|im|-----|, -1 + re|-----|||
            /    |       |   2 ||       |   2 |     |     |  |  ___|         |  ___|||          /    |       |   2 ||       |   2 |     |     |  |  ___|         |  ___|||
           /     |       |  e  ||      2|  e  |     |     \  \\/ y /         \\/ y //|         /     |       |  e  ||      2|  e  |     |     \  \\/ y /         \\/ y //|
x1 = -    /      |-1 + re|-----||  + im |-----| *cos|--------------------------------| - I*   /      |-1 + re|-----||  + im |-----| *sin|--------------------------------|
       4 /       |       |  ___||       |  ___|     \               2                /     4 /       |       |  ___||       |  ___|     \               2                /
       \/        \       \\/ y //       \\/ y /                                            \/        \       \\/ y //       \\/ y /                                       
$$x_{1} = - i \sqrt[4]{\left(\operatorname{re}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)},\operatorname{re}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)} - 1 \right)}}{2} \right)} - \sqrt[4]{\left(\operatorname{re}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)},\operatorname{re}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)} - 1 \right)}}{2} \right)}$$
                                                  /     /  /   c \         /   c \\\                                                  /     /  /   c \         /   c \\\
              ________________________________    |     |  |   - |         |   - |||              ________________________________    |     |  |   - |         |   - |||
             /                 2                  |     |  |   2 |         |   2 |||             /                 2                  |     |  |   2 |         |   2 |||
            /  /       /   c \\       /   c \     |     |  |  e  |         |  e  |||            /  /       /   c \\       /   c \     |     |  |  e  |         |  e  |||
           /   |       |   - ||       |   - |     |atan2|im|-----|, -1 + re|-----|||           /   |       |   - ||       |   - |     |atan2|im|-----|, -1 + re|-----|||
          /    |       |   2 ||       |   2 |     |     |  |  ___|         |  ___|||          /    |       |   2 ||       |   2 |     |     |  |  ___|         |  ___|||
         /     |       |  e  ||      2|  e  |     |     \  \\/ y /         \\/ y //|         /     |       |  e  ||      2|  e  |     |     \  \\/ y /         \\/ y //|
x2 =    /      |-1 + re|-----||  + im |-----| *cos|--------------------------------| + I*   /      |-1 + re|-----||  + im |-----| *sin|--------------------------------|
     4 /       |       |  ___||       |  ___|     \               2                /     4 /       |       |  ___||       |  ___|     \               2                /
     \/        \       \\/ y //       \\/ y /                                            \/        \       \\/ y //       \\/ y /                                       
$$x_{2} = i \sqrt[4]{\left(\operatorname{re}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)},\operatorname{re}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)} - 1 \right)}}{2} \right)} + \sqrt[4]{\left(\operatorname{re}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)},\operatorname{re}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)} - 1 \right)}}{2} \right)}$$
x2 = i*((re(exp(c/2)/sqrt(y)) - 1)^2 + im(exp(c/2)/sqrt(y))^2)^(1/4)*sin(atan2(im(exp(c/2)/sqrt(y), re(exp(c/2)/sqrt(y)) - 1)/2) + ((re(exp(c/2)/sqrt(y)) - 1)^2 + im(exp(c/2)/sqrt(y))^2)^(1/4)*cos(atan2(im(exp(c/2)/sqrt(y)), re(exp(c/2)/sqrt(y)) - 1)/2))
Suma y producto de raíces [src]
suma
                                               /     /  /   c \         /   c \\\                                                  /     /  /   c \         /   c \\\                                                /     /  /   c \         /   c \\\                                                  /     /  /   c \         /   c \\\
           ________________________________    |     |  |   - |         |   - |||              ________________________________    |     |  |   - |         |   - |||            ________________________________    |     |  |   - |         |   - |||              ________________________________    |     |  |   - |         |   - |||
          /                 2                  |     |  |   2 |         |   2 |||             /                 2                  |     |  |   2 |         |   2 |||           /                 2                  |     |  |   2 |         |   2 |||             /                 2                  |     |  |   2 |         |   2 |||
         /  /       /   c \\       /   c \     |     |  |  e  |         |  e  |||            /  /       /   c \\       /   c \     |     |  |  e  |         |  e  |||          /  /       /   c \\       /   c \     |     |  |  e  |         |  e  |||            /  /       /   c \\       /   c \     |     |  |  e  |         |  e  |||
        /   |       |   - ||       |   - |     |atan2|im|-----|, -1 + re|-----|||           /   |       |   - ||       |   - |     |atan2|im|-----|, -1 + re|-----|||         /   |       |   - ||       |   - |     |atan2|im|-----|, -1 + re|-----|||           /   |       |   - ||       |   - |     |atan2|im|-----|, -1 + re|-----|||
       /    |       |   2 ||       |   2 |     |     |  |  ___|         |  ___|||          /    |       |   2 ||       |   2 |     |     |  |  ___|         |  ___|||        /    |       |   2 ||       |   2 |     |     |  |  ___|         |  ___|||          /    |       |   2 ||       |   2 |     |     |  |  ___|         |  ___|||
      /     |       |  e  ||      2|  e  |     |     \  \\/ y /         \\/ y //|         /     |       |  e  ||      2|  e  |     |     \  \\/ y /         \\/ y //|       /     |       |  e  ||      2|  e  |     |     \  \\/ y /         \\/ y //|         /     |       |  e  ||      2|  e  |     |     \  \\/ y /         \\/ y //|
-    /      |-1 + re|-----||  + im |-----| *cos|--------------------------------| - I*   /      |-1 + re|-----||  + im |-----| *sin|--------------------------------| +    /      |-1 + re|-----||  + im |-----| *cos|--------------------------------| + I*   /      |-1 + re|-----||  + im |-----| *sin|--------------------------------|
  4 /       |       |  ___||       |  ___|     \               2                /     4 /       |       |  ___||       |  ___|     \               2                /   4 /       |       |  ___||       |  ___|     \               2                /     4 /       |       |  ___||       |  ___|     \               2                /
  \/        \       \\/ y //       \\/ y /                                            \/        \       \\/ y //       \\/ y /                                          \/        \       \\/ y //       \\/ y /                                            \/        \       \\/ y //       \\/ y /                                       
$$\left(- i \sqrt[4]{\left(\operatorname{re}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)},\operatorname{re}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)} - 1 \right)}}{2} \right)} - \sqrt[4]{\left(\operatorname{re}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)},\operatorname{re}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)} - 1 \right)}}{2} \right)}\right) + \left(i \sqrt[4]{\left(\operatorname{re}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)},\operatorname{re}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)} - 1 \right)}}{2} \right)} + \sqrt[4]{\left(\operatorname{re}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)},\operatorname{re}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)} - 1 \right)}}{2} \right)}\right)$$
=
0
$$0$$
producto
/                                               /     /  /   c \         /   c \\\                                                  /     /  /   c \         /   c \\\\ /                                             /     /  /   c \         /   c \\\                                                  /     /  /   c \         /   c \\\\
|           ________________________________    |     |  |   - |         |   - |||              ________________________________    |     |  |   - |         |   - |||| |         ________________________________    |     |  |   - |         |   - |||              ________________________________    |     |  |   - |         |   - ||||
|          /                 2                  |     |  |   2 |         |   2 |||             /                 2                  |     |  |   2 |         |   2 |||| |        /                 2                  |     |  |   2 |         |   2 |||             /                 2                  |     |  |   2 |         |   2 ||||
|         /  /       /   c \\       /   c \     |     |  |  e  |         |  e  |||            /  /       /   c \\       /   c \     |     |  |  e  |         |  e  |||| |       /  /       /   c \\       /   c \     |     |  |  e  |         |  e  |||            /  /       /   c \\       /   c \     |     |  |  e  |         |  e  ||||
|        /   |       |   - ||       |   - |     |atan2|im|-----|, -1 + re|-----|||           /   |       |   - ||       |   - |     |atan2|im|-----|, -1 + re|-----|||| |      /   |       |   - ||       |   - |     |atan2|im|-----|, -1 + re|-----|||           /   |       |   - ||       |   - |     |atan2|im|-----|, -1 + re|-----||||
|       /    |       |   2 ||       |   2 |     |     |  |  ___|         |  ___|||          /    |       |   2 ||       |   2 |     |     |  |  ___|         |  ___|||| |     /    |       |   2 ||       |   2 |     |     |  |  ___|         |  ___|||          /    |       |   2 ||       |   2 |     |     |  |  ___|         |  ___||||
|      /     |       |  e  ||      2|  e  |     |     \  \\/ y /         \\/ y //|         /     |       |  e  ||      2|  e  |     |     \  \\/ y /         \\/ y //|| |    /     |       |  e  ||      2|  e  |     |     \  \\/ y /         \\/ y //|         /     |       |  e  ||      2|  e  |     |     \  \\/ y /         \\/ y //||
|-    /      |-1 + re|-----||  + im |-----| *cos|--------------------------------| - I*   /      |-1 + re|-----||  + im |-----| *sin|--------------------------------||*|   /      |-1 + re|-----||  + im |-----| *cos|--------------------------------| + I*   /      |-1 + re|-----||  + im |-----| *sin|--------------------------------||
|  4 /       |       |  ___||       |  ___|     \               2                /     4 /       |       |  ___||       |  ___|     \               2                /| |4 /       |       |  ___||       |  ___|     \               2                /     4 /       |       |  ___||       |  ___|     \               2                /|
\  \/        \       \\/ y //       \\/ y /                                            \/        \       \\/ y //       \\/ y /                                       / \\/        \       \\/ y //       \\/ y /                                            \/        \       \\/ y //       \\/ y /                                       /
$$\left(- i \sqrt[4]{\left(\operatorname{re}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)},\operatorname{re}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)} - 1 \right)}}{2} \right)} - \sqrt[4]{\left(\operatorname{re}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)},\operatorname{re}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)} - 1 \right)}}{2} \right)}\right) \left(i \sqrt[4]{\left(\operatorname{re}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)},\operatorname{re}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)} - 1 \right)}}{2} \right)} + \sqrt[4]{\left(\operatorname{re}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)},\operatorname{re}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)} - 1 \right)}}{2} \right)}\right)$$
=
                                                   /  /   c \         /   c \\
          ________________________________         |  |   - |         |   - ||
         /                 2                       |  |   2 |         |   2 ||
        /  /       /   c \\       /   c \          |  |  e  |         |  e  ||
       /   |       |   - ||       |   - |   I*atan2|im|-----|, -1 + re|-----||
      /    |       |   2 ||       |   2 |          |  |  ___|         |  ___||
     /     |       |  e  ||      2|  e  |          \  \\/ y /         \\/ y //
-   /      |-1 + re|-----||  + im |-----| *e                                  
   /       |       |  ___||       |  ___|                                     
 \/        \       \\/ y //       \\/ y /                                     
$$- \sqrt{\left(\operatorname{re}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)}\right)^{2}} e^{i \operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)},\operatorname{re}{\left(\frac{e^{\frac{c}{2}}}{\sqrt{y}}\right)} - 1 \right)}}$$
-sqrt((-1 + re(exp(c/2)/sqrt(y)))^2 + im(exp(c/2)/sqrt(y))^2)*exp(i*atan2(im(exp(c/2)/sqrt(y)), -1 + re(exp(c/2)/sqrt(y))))