Solve dx+xydy=y^2dx+ydy

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

Solve for d (complex solution)

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

Steps for Solving Linear Equation

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

$$dx+xy^{2}d=y^{2}dx+ydy$$

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

$$dx+xy^{2}d=y^{2}dx+y^{2}d$$

Subtract y^{2}dx from both sides.

$$dx+xy^{2}d-y^{2}dx=y^{2}d$$

Combine xy^{2}d and -y^{2}dx to get 0.

$$dx=y^{2}d$$

Subtract y^{2}d from both sides.

$$dx-y^{2}d=0$$

Reorder the terms.

$$dx-dy^{2}=0$$

Combine all terms containing d.

$$\left(x-y^{2}\right)d=0$$

Divide 0 by x-y^{2}.

$$d=0$$

Solve for x (complex solution)

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

Steps for Solving Linear Equation

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

$$dx+xy^{2}d=y^{2}dx+ydy$$

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

$$dx+xy^{2}d=y^{2}dx+y^{2}d$$

Subtract y^{2}dx from both sides.

$$dx+xy^{2}d-y^{2}dx=y^{2}d$$

Combine xy^{2}d and -y^{2}dx to get 0.

$$dx=y^{2}d$$

The equation is in standard form.

$$dx=dy^{2}$$

Divide both sides by d.

$$\frac{dx}{d}=\frac{dy^{2}}{d}$$

Dividing by d undoes the multiplication by d.

$$x=\frac{dy^{2}}{d}$$

Divide dy^{2} by d.

$$x=y^{2}$$

Solve for d

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

Steps for Solving Linear Equation

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

$$dx+xy^{2}d=y^{2}dx+ydy$$

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

$$dx+xy^{2}d=y^{2}dx+y^{2}d$$

Subtract y^{2}dx from both sides.

$$dx+xy^{2}d-y^{2}dx=y^{2}d$$

Combine xy^{2}d and -y^{2}dx to get 0.

$$dx=y^{2}d$$

Subtract y^{2}d from both sides.

$$dx-y^{2}d=0$$

Reorder the terms.

$$dx-dy^{2}=0$$

Combine all terms containing d.

$$\left(x-y^{2}\right)d=0$$

Divide 0 by x-y^{2}.

$$d=0$$

Solve for x

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

Steps for Solving Linear Equation

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

$$dx+xy^{2}d=y^{2}dx+ydy$$

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

$$dx+xy^{2}d=y^{2}dx+y^{2}d$$

Subtract y^{2}dx from both sides.

$$dx+xy^{2}d-y^{2}dx=y^{2}d$$

Combine xy^{2}d and -y^{2}dx to get 0.

$$dx=y^{2}d$$

The equation is in standard form.

$$dx=dy^{2}$$

Divide both sides by d.

$$\frac{dx}{d}=\frac{dy^{2}}{d}$$

Dividing by d undoes the multiplication by d.

$$x=\frac{dy^{2}}{d}$$

Divide dy^{2} by d.

$$x=y^{2}$$