Solve xy(3x+6/y)dx+xy(x^2/y+3y/x)dy=0

$$x y ( 3 x + \frac { 6 } { y } ) d x + x y ( \frac { x ^ { 2 } } { y } + 3 \frac { y } { x } ) d y = 0$$

Solve for d

$$\left\{\begin{matrix}d=0\text{, }&y\neq 0\text{ and }x\neq 0\\d\in \mathrm{R}\text{, }&x\neq 0\text{ and }6x^{2}+3y^{3}+4yx^{3}=0\text{ and }y\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by xy, the least common multiple of y,x.

$$xy\left(3x+\frac{6}{y}\right)dxxy+xy\left(\frac{x^{2}}{y}+3\times \frac{y}{x}\right)dyxy=0$$

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

$$x^{2}y\left(3x+\frac{6}{y}\right)dxy+xy\left(\frac{x^{2}}{y}+3\times \frac{y}{x}\right)dyxy=0$$

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

$$x^{2}y\left(3x+\frac{6}{y}\right)dxy+xy^{2}\left(\frac{x^{2}}{y}+3\times \frac{y}{x}\right)dxy=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3x times \frac{y}{y}.

$$x^{2}y\left(\frac{3xy}{y}+\frac{6}{y}\right)dxy+xy^{2}\left(\frac{x^{2}}{y}+3\times \frac{y}{x}\right)dxy=0$$

Since \frac{3xy}{y} and \frac{6}{y} have the same denominator, add them by adding their numerators.

$$x^{2}y\times \frac{3xy+6}{y}dxy+xy^{2}\left(\frac{x^{2}}{y}+3\times \frac{y}{x}\right)dxy=0$$

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

$$x^{3}y\times \frac{3xy+6}{y}dy+xy^{2}\left(\frac{x^{2}}{y}+3\times \frac{y}{x}\right)dxy=0$$

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

$$x^{3}y^{2}\times \frac{3xy+6}{y}d+xy^{2}\left(\frac{x^{2}}{y}+3\times \frac{y}{x}\right)dxy=0$$

Express x^{3}\times \frac{3xy+6}{y} as a single fraction.

$$\frac{x^{3}\left(3xy+6\right)}{y}y^{2}d+xy^{2}\left(\frac{x^{2}}{y}+3\times \frac{y}{x}\right)dxy=0$$

Express \frac{x^{3}\left(3xy+6\right)}{y}d as a single fraction.

$$\frac{x^{3}\left(3xy+6\right)d}{y}y^{2}+xy^{2}\left(\frac{x^{2}}{y}+3\times \frac{y}{x}\right)dxy=0$$

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

$$\frac{x^{3}\left(3xy+6\right)d}{y}y^{2}+x^{2}y^{2}\left(\frac{x^{2}}{y}+3\times \frac{y}{x}\right)dy=0$$

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

$$\frac{x^{3}\left(3xy+6\right)d}{y}y^{2}+x^{2}y^{3}\left(\frac{x^{2}}{y}+3\times \frac{y}{x}\right)d=0$$

Express 3\times \frac{y}{x} as a single fraction.

$$\frac{x^{3}\left(3xy+6\right)d}{y}y^{2}+x^{2}y^{3}\left(\frac{x^{2}}{y}+\frac{3y}{x}\right)d=0$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y and x is xy. Multiply \frac{x^{2}}{y} times \frac{x}{x}. Multiply \frac{3y}{x} times \frac{y}{y}.

$$\frac{x^{3}\left(3xy+6\right)d}{y}y^{2}+x^{2}y^{3}\left(\frac{x^{2}x}{xy}+\frac{3yy}{xy}\right)d=0$$

Since \frac{x^{2}x}{xy} and \frac{3yy}{xy} have the same denominator, add them by adding their numerators.

$$\frac{x^{3}\left(3xy+6\right)d}{y}y^{2}+x^{2}y^{3}\times \frac{x^{2}x+3yy}{xy}d=0$$

Do the multiplications in x^{2}x+3yy.

$$\frac{x^{3}\left(3xy+6\right)d}{y}y^{2}+x^{2}y^{3}\times \frac{x^{3}+3y^{2}}{xy}d=0$$

Express x^{2}\times \frac{x^{3}+3y^{2}}{xy} as a single fraction.

$$\frac{x^{3}\left(3xy+6\right)d}{y}y^{2}+\frac{x^{2}\left(x^{3}+3y^{2}\right)}{xy}y^{3}d=0$$

Cancel out x in both numerator and denominator.

$$\frac{x^{3}\left(3xy+6\right)d}{y}y^{2}+\frac{x\left(x^{3}+3y^{2}\right)}{y}y^{3}d=0$$

Express \frac{x\left(x^{3}+3y^{2}\right)}{y}d as a single fraction.

$$\frac{x^{3}\left(3xy+6\right)d}{y}y^{2}+\frac{x\left(x^{3}+3y^{2}\right)d}{y}y^{3}=0$$

Use the distributive property to multiply x^{3} by 3xy+6.

$$\frac{\left(3yx^{4}+6x^{3}\right)d}{y}y^{2}+\frac{x\left(x^{3}+3y^{2}\right)d}{y}y^{3}=0$$

Use the distributive property to multiply 3yx^{4}+6x^{3} by d.

$$\frac{3yx^{4}d+6x^{3}d}{y}y^{2}+\frac{x\left(x^{3}+3y^{2}\right)d}{y}y^{3}=0$$

Express \frac{3yx^{4}d+6x^{3}d}{y}y^{2} as a single fraction.

$$\frac{\left(3yx^{4}d+6x^{3}d\right)y^{2}}{y}+\frac{x\left(x^{3}+3y^{2}\right)d}{y}y^{3}=0$$

Cancel out y in both numerator and denominator.

$$y\left(6dx^{3}+3dyx^{4}\right)+\frac{x\left(x^{3}+3y^{2}\right)d}{y}y^{3}=0$$

Use the distributive property to multiply y by 6dx^{3}+3dyx^{4}.

$$6ydx^{3}+3dx^{4}y^{2}+\frac{x\left(x^{3}+3y^{2}\right)d}{y}y^{3}=0$$

Use the distributive property to multiply x by x^{3}+3y^{2}.

$$6ydx^{3}+3dx^{4}y^{2}+\frac{\left(x^{4}+3xy^{2}\right)d}{y}y^{3}=0$$

Use the distributive property to multiply x^{4}+3xy^{2} by d.

$$6ydx^{3}+3dx^{4}y^{2}+\frac{x^{4}d+3xy^{2}d}{y}y^{3}=0$$

Express \frac{x^{4}d+3xy^{2}d}{y}y^{3} as a single fraction.

$$6ydx^{3}+3dx^{4}y^{2}+\frac{\left(x^{4}d+3xy^{2}d\right)y^{3}}{y}=0$$

Cancel out y in both numerator and denominator.

$$6ydx^{3}+3dx^{4}y^{2}+y^{2}\left(dx^{4}+3dxy^{2}\right)=0$$

Use the distributive property to multiply y^{2} by dx^{4}+3dxy^{2}.

$$6ydx^{3}+3dx^{4}y^{2}+y^{2}dx^{4}+3dxy^{4}=0$$

Combine 3dx^{4}y^{2} and y^{2}dx^{4} to get 4dx^{4}y^{2}.

$$6ydx^{3}+4dx^{4}y^{2}+3dxy^{4}=0$$

Combine all terms containing d.

$$\left(6yx^{3}+4x^{4}y^{2}+3xy^{4}\right)d=0$$

The equation is in standard form.

$$\left(3xy^{4}+4y^{2}x^{4}+6yx^{3}\right)d=0$$

Divide 0 by 6yx^{3}+4x^{4}y^{2}+3xy^{4}.

$$d=0$$