Solve dy/dx=xdx+y

$$\frac { d y } { d x } = x d x + y$$

Solve for d (complex solution)

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

Steps for Solving Linear Equation

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

$$\frac{\mathrm{d}(y)}{\mathrm{d}x}=x^{2}d+y$$

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

$$x^{2}d+y=\frac{\mathrm{d}(y)}{\mathrm{d}x}$$

Subtract y from both sides.

$$x^{2}d=\frac{\mathrm{d}(y)}{\mathrm{d}x}-y$$

The equation is in standard form.

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

Divide both sides by x^{2}.

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

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

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

Solve for d

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

Steps for Solving Linear Equation

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

$$\frac{\mathrm{d}(y)}{\mathrm{d}x}=x^{2}d+y$$

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

$$x^{2}d+y=\frac{\mathrm{d}(y)}{\mathrm{d}x}$$

Subtract y from both sides.

$$x^{2}d=\frac{\mathrm{d}(y)}{\mathrm{d}x}-y$$

The equation is in standard form.

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

Divide both sides by x^{2}.

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

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

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

Solve for x (complex solution)

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

Solve for x

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