Solve sqrt{x}dy=sqrt{y}dx

$$\sqrt { x } d y = \sqrt { y } d x$$

Solve for d (complex solution)

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

Steps for Solving Linear Equation

Subtract \sqrt{y}dx from both sides.

$$\sqrt{x}dy-\sqrt{y}dx=0$$

Combine all terms containing d.

$$\left(\sqrt{x}y-\sqrt{y}x\right)d=0$$

Divide 0 by \sqrt{x}y-\sqrt{y}x.

$$d=0$$

Solve for d

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

Steps for Solving Linear Equation

Subtract \sqrt{y}dx from both sides.

$$\sqrt{x}dy-\sqrt{y}dx=0$$

Combine all terms containing d.

$$\left(\sqrt{x}y-\sqrt{y}x\right)d=0$$

Divide 0 by \sqrt{x}y-\sqrt{y}x.

$$d=0$$

Solve for x (complex solution)

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

Solve for x

$$\left\{\begin{matrix}x=y\text{; }x=0\text{, }&y\geq 0\\x\geq 0\text{, }&y=0\text{ or }\left(d=0\text{ and }y\geq 0\right)\end{matrix}\right.$$