Solve fx^2dx-(xdy+ydx)+ydy=0

$$f x ^ { 2 } d x - ( x d y + y d x ) + y d y = 0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&\left(x=0\text{ and }y=0\right)\text{ or }\left(f=-\frac{y\left(y-2x\right)}{x^{3}}\text{ and }x\neq 0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$fx^{3}d-\left(xdy+ydx\right)+ydy=0$$

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

$$fx^{3}d-\left(xdy+ydx\right)+y^{2}d=0$$

Combine xdy and ydx to get 2xdy.

$$fx^{3}d-2xdy+y^{2}d=0$$

Combine all terms containing d.

$$\left(fx^{3}-2xy+y^{2}\right)d=0$$

Divide 0 by fx^{3}-2xy+y^{2}.

$$d=0$$

Solve for f

$$\left\{\begin{matrix}f=-\frac{y\left(y-2x\right)}{x^{3}}\text{, }&x\neq 0\\f\in \mathrm{R}\text{, }&\left(y=0\text{ and }x=0\right)\text{ or }d=0\end{matrix}\right.$$

Steps for Solving Linear Equation

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$fx^{3}d-\left(xdy+ydx\right)+ydy=0$$

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

$$fx^{3}d-\left(xdy+ydx\right)+y^{2}d=0$$

Combine xdy and ydx to get 2xdy.

$$fx^{3}d-2xdy+y^{2}d=0$$

Add 2xdy to both sides. Anything plus zero gives itself.

$$fx^{3}d+y^{2}d=2xdy$$

Subtract y^{2}d from both sides.

$$fx^{3}d=2xdy-y^{2}d$$

Reorder the terms.

$$dfx^{3}=2dxy-dy^{2}$$

The equation is in standard form.

$$dx^{3}f=2dxy-dy^{2}$$

Divide both sides by dx^{3}.

$$\frac{dx^{3}f}{dx^{3}}=\frac{dy\left(2x-y\right)}{dx^{3}}$$

Dividing by dx^{3} undoes the multiplication by dx^{3}.

$$f=\frac{dy\left(2x-y\right)}{dx^{3}}$$

Divide dy\left(2x-y\right) by dx^{3}.

$$f=\frac{y\left(2x-y\right)}{x^{3}}$$