Solve xdx-y^2dy=0

$$x d x - y ^ { 2 } d y = 0$$

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&x=-y^{\frac{3}{2}}\text{ or }x=y^{\frac{3}{2}}\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$x^{2}d-y^{2}dy=0$$

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

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

Reorder the terms.

$$dx^{2}-dy^{3}=0$$

Combine all terms containing d.

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

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

$$d=0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&y=x^{\frac{2}{3}}\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$x^{2}d-y^{2}dy=0$$

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

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

Reorder the terms.

$$dx^{2}-dy^{3}=0$$

Combine all terms containing d.

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

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

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=-y^{\frac{3}{2}}\text{; }x=y^{\frac{3}{2}}\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&d=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=y^{\frac{3}{2}}\text{; }x=-y^{\frac{3}{2}}\text{, }&y\geq 0\\x\in \mathrm{R}\text{, }&d=0\end{matrix}\right.$$