(-y - 1)*exp(-y) = c - x^2/2 (уравнение)

Найду корень уравнения: (-y - 1)*exp(-y) = c - x^2/2

Вы ввели

                    2
          -y       x 
(-y - 1)*e   = c - --
                   2 

$$\left(- y - 1\right) e^{- y} = c - \frac{x^{2}}{2}$$

Подробное решение

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

Уравнение превратится из
$$\left(- y - 1\right) e^{- y} = c - \frac{x^{2}}{2}$$
в
$$\left(- c + \frac{x^{2}}{2}\right) + \left(- y - 1\right) e^{- y} = 0$$
Раскроем выражение в уравнении
$$\left(- c + \frac{x^{2}}{2}\right) + \left(- y - 1\right) e^{- y} = 0$$
Получаем квадратное уравнение
$$- c + \frac{x^{2}}{2} - y e^{- y} - e^{- y} = 0$$
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.
$$a = \frac{1}{2}$$
$$b = 0$$
$$c = - c - y e^{- y} - e^{- y}$$
, то

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

(0)^2 - 4 * (1/2) * (-c - exp(-y) - y*exp(-y)) = 2*c + 2*exp(-y) + 2*y*exp(-y)

Уравнение имеет два корня.

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

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

или
$$x_{1} = \sqrt{2 c + 2 y e^{- y} + 2 e^{- y}}$$
$$x_{2} = - \sqrt{2 c + 2 y e^{- y} + 2 e^{- y}}$$

Быстрый ответ

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                /                                                                                    2                                                                                      2     /     //          /   y\\             -re(y)   /              /   y\\  -re(y)             /          /   y\\  -re(y)              /              /   y\\             -re(y)\\              /                                                                                    2                                                                                      2     /     //          /   y\\             -re(y)   /              /   y\\  -re(y)             /          /   y\\  -re(y)              /              /   y\\             -re(y)\\
         ___ 4 /  //          /   y\\             -re(y)   /              /   y\\  -re(y)           \    //          /   y\\  -re(y)              /              /   y\\             -re(y)\      |atan2\\im(y) + im\c*e //*cos(im(y))*e       - \1 + re(y) + re\c*e //*e      *sin(im(y)), \im(y) + im\c*e //*e      *sin(im(y)) + \1 + re(y) + re\c*e //*cos(im(y))*e      /|       ___ 4 /  //          /   y\\             -re(y)   /              /   y\\  -re(y)           \    //          /   y\\  -re(y)              /              /   y\\             -re(y)\      |atan2\\im(y) + im\c*e //*cos(im(y))*e       - \1 + re(y) + re\c*e //*e      *sin(im(y)), \im(y) + im\c*e //*e      *sin(im(y)) + \1 + re(y) + re\c*e //*cos(im(y))*e      /|
x1 = - \/ 2 *\/   \\im(y) + im\c*e //*cos(im(y))*e       - \1 + re(y) + re\c*e //*e      *sin(im(y))/  + \\im(y) + im\c*e //*e      *sin(im(y)) + \1 + re(y) + re\c*e //*cos(im(y))*e      /  *cos|---------------------------------------------------------------------------------------------------------------------------------------------------------------------------| - I*\/ 2 *\/   \\im(y) + im\c*e //*cos(im(y))*e       - \1 + re(y) + re\c*e //*e      *sin(im(y))/  + \\im(y) + im\c*e //*e      *sin(im(y)) + \1 + re(y) + re\c*e //*cos(im(y))*e      /  *sin|---------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
                                                                                                                                                                                                  \                                                                                     2                                                                                     /                                                                                                                                                                                                \                                                                                     2                                                                                     /

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

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              /                                                                                    2                                                                                      2     /     //          /   y\\             -re(y)   /              /   y\\  -re(y)             /          /   y\\  -re(y)              /              /   y\\             -re(y)\\              /                                                                                    2                                                                                      2     /     //          /   y\\             -re(y)   /              /   y\\  -re(y)             /          /   y\\  -re(y)              /              /   y\\             -re(y)\\
       ___ 4 /  //          /   y\\             -re(y)   /              /   y\\  -re(y)           \    //          /   y\\  -re(y)              /              /   y\\             -re(y)\      |atan2\\im(y) + im\c*e //*cos(im(y))*e       - \1 + re(y) + re\c*e //*e      *sin(im(y)), \im(y) + im\c*e //*e      *sin(im(y)) + \1 + re(y) + re\c*e //*cos(im(y))*e      /|       ___ 4 /  //          /   y\\             -re(y)   /              /   y\\  -re(y)           \    //          /   y\\  -re(y)              /              /   y\\             -re(y)\      |atan2\\im(y) + im\c*e //*cos(im(y))*e       - \1 + re(y) + re\c*e //*e      *sin(im(y)), \im(y) + im\c*e //*e      *sin(im(y)) + \1 + re(y) + re\c*e //*cos(im(y))*e      /|
x2 = \/ 2 *\/   \\im(y) + im\c*e //*cos(im(y))*e       - \1 + re(y) + re\c*e //*e      *sin(im(y))/  + \\im(y) + im\c*e //*e      *sin(im(y)) + \1 + re(y) + re\c*e //*cos(im(y))*e      /  *cos|---------------------------------------------------------------------------------------------------------------------------------------------------------------------------| + I*\/ 2 *\/   \\im(y) + im\c*e //*cos(im(y))*e       - \1 + re(y) + re\c*e //*e      *sin(im(y))/  + \\im(y) + im\c*e //*e      *sin(im(y)) + \1 + re(y) + re\c*e //*cos(im(y))*e      /  *sin|---------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
                                                                                                                                                                                                \                                                                                     2                                                                                     /                                                                                                                                                                                                \                                                                                     2                                                                                     /

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