Solve ∫ y wrt y=xdx

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

Solve for d (complex solution)

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

Steps for Solving Linear Equation

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

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

Swap sides so that all variable terms are on the left hand side.

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

The equation is in standard form.

$$x^{2}d=\frac{y^{2}}{2}+С$$

Divide both sides by x^{2}.

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

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

$$d=\frac{\frac{y^{2}}{2}+С}{x^{2}}$$

Divide \frac{y^{2}}{2}+С by x^{2}.

$$d=\frac{y^{2}+2С}{2x^{2}}$$

Solve for d

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

Steps for Solving Linear Equation

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

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

Swap sides so that all variable terms are on the left hand side.

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

The equation is in standard form.

$$x^{2}d=\frac{y^{2}}{2}+С$$

Divide both sides by x^{2}.

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

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

$$d=\frac{\frac{y^{2}}{2}+С}{x^{2}}$$

Divide \frac{y^{2}}{2}+С by x^{2}.

$$d=\frac{y^{2}+2С}{2x^{2}}$$

Solve for x (complex solution)

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

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{2\times \frac{y^{2}-С}{d}}}{2}\text{; }x=-\frac{\sqrt{2\times \frac{y^{2}-С}{d}}}{2}\text{, }&\left(|y|\geq \sqrt{С}\text{ and }d>0\right)\text{ or }\left(С_{1}\geq 0\text{ and }d\neq 0\text{ and }|y|=\sqrt{С_{2}}\right)\text{ or }\left(d>0\text{ and }С_{3}<0\right)\text{ or }\left(С_{4}\geq 0\text{ and }|y|\leq \sqrt{С_{5}}\text{ and }d<0\right)\\x\in \mathrm{R}\text{, }&С=-\frac{y^{2}}{2}\text{ and }d=0\end{matrix}\right.$$