xyy=1-x^2 (уравнение)

Найду корень уравнения: xyy=1-x^2

Вы ввели

             2
x*y*y = 1 - x 

$$y x y = 1 - x^{2}$$

Подробное решение

Перенесём правую часть уравнения в
левую часть уравнения со знаком минус.

Уравнение превратится из
$$y x y = 1 - x^{2}$$
в
$$y x y + \left(x^{2} - 1\right) = 0$$
Это уравнение вида

a*x^2 + b*x + c = 0

Квадратное уравнение можно решить
с помощью дискриминанта.
Корни квадратного уравнения:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
где D = b^2 - 4*a*c - это дискриминант.
Т.к.
$$a = 1$$
$$b = y^{2}$$
$$c = -1$$
, то

D = b^2 - 4 * a * c = 

(y^2)^2 - 4 * (1) * (-1) = 4 + y^4

Уравнение имеет два корня.

x1 = (-b + sqrt(D)) / (2*a)

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

или
$$x_{1} = - \frac{y^{2}}{2} + \frac{\sqrt{y^{4} + 4}}{2}$$
$$x_{2} = - \frac{y^{2}}{2} - \frac{\sqrt{y^{4} + 4}}{2}$$

Быстрый ответ

                         /                   _________________________________________________________________________________                                                                                     \       _________________________________________________________________________________                                                                                     
                         |                  /                                    2                                          2     /     /      3                3                 4        4          2      2   \\|      /                                    2                                          2     /     /      3                3                 4        4          2      2   \\
                         |               4 /  /      3                3         \    /      4        4          2      2   \      |atan2\- 4*im (y)*re(y) + 4*re (y)*im(y), 4 + im (y) + re (y) - 6*im (y)*re (y)/||   4 /  /      3                3         \    /      4        4          2      2   \      |atan2\- 4*im (y)*re(y) + 4*re (y)*im(y), 4 + im (y) + re (y) - 6*im (y)*re (y)/|
       2        2        |               \/   \- 4*im (y)*re(y) + 4*re (y)*im(y)/  + \4 + im (y) + re (y) - 6*im (y)*re (y)/  *sin|-------------------------------------------------------------------------------||   \/   \- 4*im (y)*re(y) + 4*re (y)*im(y)/  + \4 + im (y) + re (y) - 6*im (y)*re (y)/  *cos|-------------------------------------------------------------------------------|
     im (y)   re (y)     |                                                                                                        \                                       2                                       /|                                                                                            \                                       2                                       /
x1 = ------ - ------ + I*|-im(y)*re(y) - --------------------------------------------------------------------------------------------------------------------------------------------------------------------------| - --------------------------------------------------------------------------------------------------------------------------------------------------------------------------
       2        2        \                                                                                                   2                                                                                     /                                                                                       2                                                                                     

$$x_{1} = i \left(- \frac{\sqrt[4]{\left(4 \left(\operatorname{re}{\left(y\right)}\right)^{3} \operatorname{im}{\left(y\right)} - 4 \operatorname{re}{\left(y\right)} \left(\operatorname{im}{\left(y\right)}\right)^{3}\right)^{2} + \left(\left(\operatorname{re}{\left(y\right)}\right)^{4} - 6 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{4} + 4\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \left(\operatorname{re}{\left(y\right)}\right)^{3} \operatorname{im}{\left(y\right)} - 4 \operatorname{re}{\left(y\right)} \left(\operatorname{im}{\left(y\right)}\right)^{3},\left(\operatorname{re}{\left(y\right)}\right)^{4} - 6 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{4} + 4 \right)}}{2} \right)}}{2} - \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)}\right) - \frac{\sqrt[4]{\left(4 \left(\operatorname{re}{\left(y\right)}\right)^{3} \operatorname{im}{\left(y\right)} - 4 \operatorname{re}{\left(y\right)} \left(\operatorname{im}{\left(y\right)}\right)^{3}\right)^{2} + \left(\left(\operatorname{re}{\left(y\right)}\right)^{4} - 6 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{4} + 4\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \left(\operatorname{re}{\left(y\right)}\right)^{3} \operatorname{im}{\left(y\right)} - 4 \operatorname{re}{\left(y\right)} \left(\operatorname{im}{\left(y\right)}\right)^{3},\left(\operatorname{re}{\left(y\right)}\right)^{4} - 6 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{4} + 4 \right)}}{2} \right)}}{2} - \frac{\left(\operatorname{re}{\left(y\right)}\right)^{2}}{2} + \frac{\left(\operatorname{im}{\left(y\right)}\right)^{2}}{2}$$

                         /    _________________________________________________________________________________                                                                                                   \       _________________________________________________________________________________                                                                                     
                         |   /                                    2                                          2     /     /      3                3                 4        4          2      2   \\              |      /                                    2                                          2     /     /      3                3                 4        4          2      2   \\
                         |4 /  /      3                3         \    /      4        4          2      2   \      |atan2\- 4*im (y)*re(y) + 4*re (y)*im(y), 4 + im (y) + re (y) - 6*im (y)*re (y)/|              |   4 /  /      3                3         \    /      4        4          2      2   \      |atan2\- 4*im (y)*re(y) + 4*re (y)*im(y), 4 + im (y) + re (y) - 6*im (y)*re (y)/|
       2        2        |\/   \- 4*im (y)*re(y) + 4*re (y)*im(y)/  + \4 + im (y) + re (y) - 6*im (y)*re (y)/  *sin|-------------------------------------------------------------------------------|              |   \/   \- 4*im (y)*re(y) + 4*re (y)*im(y)/  + \4 + im (y) + re (y) - 6*im (y)*re (y)/  *cos|-------------------------------------------------------------------------------|
     im (y)   re (y)     |                                                                                         \                                       2                                       /              |                                                                                            \                                       2                                       /
x2 = ------ - ------ + I*|-------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - im(y)*re(y)| + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------
       2        2        \                                                                                    2                                                                                                   /                                                                                       2                                                                                     

$$x_{2} = i \left(\frac{\sqrt[4]{\left(4 \left(\operatorname{re}{\left(y\right)}\right)^{3} \operatorname{im}{\left(y\right)} - 4 \operatorname{re}{\left(y\right)} \left(\operatorname{im}{\left(y\right)}\right)^{3}\right)^{2} + \left(\left(\operatorname{re}{\left(y\right)}\right)^{4} - 6 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{4} + 4\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \left(\operatorname{re}{\left(y\right)}\right)^{3} \operatorname{im}{\left(y\right)} - 4 \operatorname{re}{\left(y\right)} \left(\operatorname{im}{\left(y\right)}\right)^{3},\left(\operatorname{re}{\left(y\right)}\right)^{4} - 6 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{4} + 4 \right)}}{2} \right)}}{2} - \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)}\right) + \frac{\sqrt[4]{\left(4 \left(\operatorname{re}{\left(y\right)}\right)^{3} \operatorname{im}{\left(y\right)} - 4 \operatorname{re}{\left(y\right)} \left(\operatorname{im}{\left(y\right)}\right)^{3}\right)^{2} + \left(\left(\operatorname{re}{\left(y\right)}\right)^{4} - 6 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{4} + 4\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \left(\operatorname{re}{\left(y\right)}\right)^{3} \operatorname{im}{\left(y\right)} - 4 \operatorname{re}{\left(y\right)} \left(\operatorname{im}{\left(y\right)}\right)^{3},\left(\operatorname{re}{\left(y\right)}\right)^{4} - 6 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{4} + 4 \right)}}{2} \right)}}{2} - \frac{\left(\operatorname{re}{\left(y\right)}\right)^{2}}{2} + \frac{\left(\operatorname{im}{\left(y\right)}\right)^{2}}{2}$$