Solve y wrt y = ∫ -xdx

$$y d y = \int - x d x$$

Solve for d (complex solution)

$$\left\{\begin{matrix}d=-\frac{\left(\frac{x}{y}\right)^{2}}{2}+\frac{С}{y^{2}}\text{, }&y\neq 0\\d\in \mathrm{C}\text{, }&С=-\frac{x^{2}}{2}\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply y and y to get y^{2}.

$$y^{2}d=\int -x\mathrm{d}x$$

The equation is in standard form.

$$y^{2}d=-\frac{x^{2}}{2}+С$$

Divide both sides by y^{2}.

$$\frac{y^{2}d}{y^{2}}=\frac{-\frac{x^{2}}{2}+С}{y^{2}}$$

Dividing by y^{2} undoes the multiplication by y^{2}.

$$d=\frac{-\frac{x^{2}}{2}+С}{y^{2}}$$

Solve for d

$$\left\{\begin{matrix}d=-\frac{\left(\frac{x}{y}\right)^{2}}{2}+\frac{С}{y^{2}}\text{, }&y\neq 0\\d\in \mathrm{R}\text{, }&С=-\frac{x^{2}}{2}\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply y and y to get y^{2}.

$$y^{2}d=\int -x\mathrm{d}x$$

The equation is in standard form.

$$y^{2}d=-\frac{x^{2}}{2}+С$$

Divide both sides by y^{2}.

$$\frac{y^{2}d}{y^{2}}=\frac{-\frac{x^{2}}{2}+С}{y^{2}}$$

Dividing by y^{2} undoes the multiplication by y^{2}.

$$d=\frac{-\frac{x^{2}}{2}+С}{y^{2}}$$

Solve for y (complex solution)

$$\left\{\begin{matrix}y=-\frac{id^{-\frac{1}{2}}\sqrt{2x^{2}+С}}{2}\text{; }y=\frac{id^{-\frac{1}{2}}\sqrt{2x^{2}+С}}{2}\text{, }&d\neq 0\\y\in \mathrm{C}\text{, }&С=-\frac{x^{2}}{2}\text{ and }d=0\end{matrix}\right.$$

Solve for y

$$\left\{\begin{matrix}y=\frac{\sqrt{\frac{4С-2x^{2}}{d}}}{2}\text{; }y=-\frac{\sqrt{\frac{4С-2x^{2}}{d}}}{2}\text{, }&\left(d<0\text{ or }С\geq \frac{x^{2}}{2}\right)\text{ and }\left(d>0\text{ or }С_{1}\leq \frac{x^{2}}{2}\right)\text{ and }d\neq 0\\y\in \mathrm{R}\text{, }&С=-\frac{x^{2}}{2}\text{ and }d=0\end{matrix}\right.$$