Solve 3ydx-2xdy+x^2y^-1(10ydx-6xdy)=0

$$3 y d x - 2 x d y + x ^ { 2 } y ^ { - 1 } ( 10 y d x - 6 x d y ) = 0$$

Solve for d (complex solution)

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

Steps for Solving Linear Equation

Combine 3ydx and -2xdy to get ydx.

$$ydx+x^{2}y^{-1}\left(10ydx-6xdy\right)=0$$

Combine 10ydx and -6xdy to get 4ydx.

$$ydx+x^{2}y^{-1}\times 4ydx=0$$

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

$$ydx+x^{3}y^{-1}\times 4yd=0$$

Reorder the terms.

$$4\times \frac{1}{y}dyx^{3}+dxy=0$$

Multiply both sides of the equation by y.

$$4\times 1dyx^{3}+dxyy=0$$

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

$$4\times 1dyx^{3}+dxy^{2}=0$$

Multiply 4 and 1 to get 4.

$$4dyx^{3}+dxy^{2}=0$$

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$

Solve for d

$$\left\{\begin{matrix}d=0\text{, }&y\neq 0\\d\in \mathrm{R}\text{, }&\left(x=0\text{ or }y=-4x^{2}\right)\text{ and }y\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Combine 3ydx and -2xdy to get ydx.

$$ydx+x^{2}y^{-1}\left(10ydx-6xdy\right)=0$$

Combine 10ydx and -6xdy to get 4ydx.

$$ydx+x^{2}y^{-1}\times 4ydx=0$$

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

$$ydx+x^{3}y^{-1}\times 4yd=0$$

Reorder the terms.

$$4\times \frac{1}{y}dyx^{3}+dxy=0$$

Multiply both sides of the equation by y.

$$4\times 1dyx^{3}+dxyy=0$$

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

$$4\times 1dyx^{3}+dxy^{2}=0$$

Multiply 4 and 1 to get 4.

$$4dyx^{3}+dxy^{2}=0$$

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{i\sqrt{y}}{2}\text{; }x=0\text{; }x=-\frac{i\sqrt{y}}{2}\text{, }&y\neq 0\\x\in \mathrm{C}\text{, }&d=0\text{ and }y\neq 0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=0\text{, }&y\neq 0\\x=\frac{\sqrt{-y}}{2}\text{; }x=-\frac{\sqrt{-y}}{2}\text{, }&y<0\\x\in \mathrm{R}\text{, }&d=0\text{ and }y\neq 0\end{matrix}\right.$$