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

$$3 x ( x y - 2 ) d x + ( x ^ { 3 } - 2 y ) d y = 0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&x=\frac{\sqrt[3]{2\sqrt{y^{2}+\frac{1}{y^{2}}}+2y+\frac{1}{y^{3}}}}{2}+\frac{\sqrt[3]{-2\sqrt{y^{2}+\frac{1}{y^{2}}}+2y+\frac{1}{y^{3}}}}{2}+\frac{1}{2y}\text{ and }y\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

Use the distributive property to multiply 3x^{2} by xy-2.

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

Use the distributive property to multiply 3yx^{3}-6x^{2} by d.

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

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

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

Use the distributive property to multiply x^{3}d-2yd by y.

$$3yx^{3}d-6x^{2}d+x^{3}dy-2dy^{2}=0$$

Combine 3yx^{3}d and x^{3}dy to get 4yx^{3}d.

$$4yx^{3}d-6x^{2}d-2dy^{2}=0$$

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$

Solve for x

$$\left\{\begin{matrix}\\x=\frac{\sqrt[3]{2\sqrt{y^{4}+1}y^{3}+2|y|y^{4}+|y|}+\sqrt[3]{-2\sqrt{y^{4}+1}y^{3}+2|y|y^{4}+|y|}+\sqrt[3]{|y|}}{2\sqrt[3]{|y|}y}\text{, }&\text{unconditionally}\\x\in \mathrm{R}\text{, }&d=0\end{matrix}\right.$$