Solve (7y^6+2x^5y)dx-(6xy^5+x^6)dy=0

$$( 7 y ^ { 6 } + 2 x ^ { 5 } y ) d x - ( 6 x y ^ { 5 } + x ^ { 6 } ) d y = 0$$

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&\left(x+y\right)\left(x^{4}+x^{2}y^{2}-xy^{3}+y^{4}-yx^{3}\right)=0\text{ or }y=0\text{ or }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 7y^{6}+2x^{5}y by d.

$$\left(7y^{6}d+2x^{5}yd\right)x-\left(6xy^{5}+x^{6}\right)dy=0$$

Use the distributive property to multiply 7y^{6}d+2x^{5}yd by x.

$$7y^{6}dx+2ydx^{6}-\left(6xy^{5}+x^{6}\right)dy=0$$

Use the distributive property to multiply 6xy^{5}+x^{6} by d.

$$7y^{6}dx+2ydx^{6}-\left(6xy^{5}d+x^{6}d\right)y=0$$

Use the distributive property to multiply 6xy^{5}d+x^{6}d by y.

$$7y^{6}dx+2ydx^{6}-\left(6xdy^{6}+x^{6}dy\right)=0$$

To find the opposite of 6xdy^{6}+x^{6}dy, find the opposite of each term.

$$7y^{6}dx+2ydx^{6}-6xdy^{6}-x^{6}dy=0$$

Combine 7y^{6}dx and -6xdy^{6} to get y^{6}dx.

$$y^{6}dx+2ydx^{6}-x^{6}dy=0$$

Combine 2ydx^{6} and -x^{6}dy to get ydx^{6}.

$$y^{6}dx+ydx^{6}=0$$

Combine all terms containing d.

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

The equation is in standard form.

$$\left(xy^{6}+yx^{6}\right)d=0$$

Divide 0 by y^{6}x+yx^{6}.

$$d=0$$

Solve for d

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

Steps for Solving Linear Equation

Use the distributive property to multiply 7y^{6}+2x^{5}y by d.

$$\left(7y^{6}d+2x^{5}yd\right)x-\left(6xy^{5}+x^{6}\right)dy=0$$

Use the distributive property to multiply 7y^{6}d+2x^{5}yd by x.

$$7y^{6}dx+2ydx^{6}-\left(6xy^{5}+x^{6}\right)dy=0$$

Use the distributive property to multiply 6xy^{5}+x^{6} by d.

$$7y^{6}dx+2ydx^{6}-\left(6xy^{5}d+x^{6}d\right)y=0$$

Use the distributive property to multiply 6xy^{5}d+x^{6}d by y.

$$7y^{6}dx+2ydx^{6}-\left(6xdy^{6}+x^{6}dy\right)=0$$

To find the opposite of 6xdy^{6}+x^{6}dy, find the opposite of each term.

$$7y^{6}dx+2ydx^{6}-6xdy^{6}-x^{6}dy=0$$

Combine 7y^{6}dx and -6xdy^{6} to get y^{6}dx.

$$y^{6}dx+2ydx^{6}-x^{6}dy=0$$

Combine 2ydx^{6} and -x^{6}dy to get ydx^{6}.

$$y^{6}dx+ydx^{6}=0$$

Combine all terms containing d.

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

The equation is in standard form.

$$\left(xy^{6}+yx^{6}\right)d=0$$

Divide 0 by y^{6}x+yx^{6}.

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x\in e^{\frac{\pi i}{5}}y,0,e^{\frac{3\pi i}{5}}y,-y,e^{\frac{7\pi i}{5}}y,e^{\frac{9\pi i}{5}}y\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&d=0\text{ or }y=0\end{matrix}\right.$$

Solve for x

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