Решение уравнений/ Любые/ xyy=1-x^2

xyy=1-x^2 (уравнение)

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    Найду корень уравнения: xyy=1-x^2

    Решение

    Вы ввели [src]
                 2
x*y*y = 1 - x
    yxy=1−x2y x y = 1 - x^{2}
    Упростить
    Подробное решение
    Перенесём правую часть уравнения в
    левую часть уравнения со знаком минус.

    Уравнение превратится из
    yxy=1−x2y x y = 1 - x^{2}
    в
    yxy+(x2−1)=0y 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:
    x1=D−b2ax_{1} = \frac{\sqrt{D} - b}{2 a}
    x2=−D−b2ax_{2} = \frac{- \sqrt{D} - b}{2 a}
    где D = b^2 - 4*a*c - это дискриминант.
    Т.к.
    a=1a = 1
    b=y2b = y^{2}
    c=−1c = -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
    x1=−y22+y4+42x_{1} = - \frac{y^{2}}{2} + \frac{\sqrt{y^{4} + 4}}{2}
    x2=−y22−y4+42x_{2} = - \frac{y^{2}}{2} - \frac{\sqrt{y^{4} + 4}}{2}
    График
    Быстрый ответ [src]
                             /                   _________________________________________________________________________________                                                                                     \       _________________________________________________________________________________                                                                                     
                         |                  /                                    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
    x1=i(−(4(re⁡(y))3im⁡(y)−4re⁡(y)(im⁡(y))3)2+((re⁡(y))4−6(re⁡(y))2(im⁡(y))2+(im⁡(y))4+4)24sin⁡(atan2⁡(4(re⁡(y))3im⁡(y)−4re⁡(y)(im⁡(y))3,(re⁡(y))4−6(re⁡(y))2(im⁡(y))2+(im⁡(y))4+4)2)2−re⁡(y)im⁡(y))−(4(re⁡(y))3im⁡(y)−4re⁡(y)(im⁡(y))3)2+((re⁡(y))4−6(re⁡(y))2(im⁡(y))2+(im⁡(y))4+4)24cos⁡(atan2⁡(4(re⁡(y))3im⁡(y)−4re⁡(y)(im⁡(y))3,(re⁡(y))4−6(re⁡(y))2(im⁡(y))2+(im⁡(y))4+4)2)2−(re⁡(y))22+(im⁡(y))22x_{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
    x2=i((4(re⁡(y))3im⁡(y)−4re⁡(y)(im⁡(y))3)2+((re⁡(y))4−6(re⁡(y))2(im⁡(y))2+(im⁡(y))4+4)24sin⁡(atan2⁡(4(re⁡(y))3im⁡(y)−4re⁡(y)(im⁡(y))3,(re⁡(y))4−6(re⁡(y))2(im⁡(y))2+(im⁡(y))4+4)2)2−re⁡(y)im⁡(y))+(4(re⁡(y))3im⁡(y)−4re⁡(y)(im⁡(y))3)2+((re⁡(y))4−6(re⁡(y))2(im⁡(y))2+(im⁡(y))4+4)24cos⁡(atan2⁡(4(re⁡(y))3im⁡(y)−4re⁡(y)(im⁡(y))3,(re⁡(y))4−6(re⁡(y))2(im⁡(y))2+(im⁡(y))4+4)2)2−(re⁡(y))22+(im⁡(y))22x_{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}