Solve dv=partialv/partialxdx+partialv/partialydy

$$d v = \frac { \partial v } { \partial x } d x + \frac { \partial v } { \partial y } d y$$

Solve for d (complex solution)

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

Steps for Solving Linear Equation

Subtract \frac{\mathrm{d}(v)}{\mathrm{d}x}dx from both sides.

$$dv-\frac{\mathrm{d}(v)}{\mathrm{d}x}dx=\frac{\mathrm{d}(v)}{\mathrm{d}y}dy$$

Subtract \frac{\mathrm{d}(v)}{\mathrm{d}y}dy from both sides.

$$dv-\frac{\mathrm{d}(v)}{\mathrm{d}x}dx-\frac{\mathrm{d}(v)}{\mathrm{d}y}dy=0$$

Reorder the terms.

$$-dx\frac{\mathrm{d}(v)}{\mathrm{d}x}-dy\frac{\mathrm{d}(v)}{\mathrm{d}y}+dv=0$$

Combine all terms containing d.

$$\left(-x\frac{\mathrm{d}(v)}{\mathrm{d}x}-y\frac{\mathrm{d}(v)}{\mathrm{d}y}+v\right)d=0$$

The equation is in standard form.

$$vd=0$$

Divide 0 by v.

$$d=0$$

Solve for d

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

Steps for Solving Linear Equation

Subtract \frac{\mathrm{d}(v)}{\mathrm{d}x}dx from both sides.

$$dv-\frac{\mathrm{d}(v)}{\mathrm{d}x}dx=\frac{\mathrm{d}(v)}{\mathrm{d}y}dy$$

Subtract \frac{\mathrm{d}(v)}{\mathrm{d}y}dy from both sides.

$$dv-\frac{\mathrm{d}(v)}{\mathrm{d}x}dx-\frac{\mathrm{d}(v)}{\mathrm{d}y}dy=0$$

Reorder the terms.

$$-dx\frac{\mathrm{d}(v)}{\mathrm{d}x}-dy\frac{\mathrm{d}(v)}{\mathrm{d}y}+dv=0$$

Combine all terms containing d.

$$\left(-x\frac{\mathrm{d}(v)}{\mathrm{d}x}-y\frac{\mathrm{d}(v)}{\mathrm{d}y}+v\right)d=0$$

The equation is in standard form.

$$vd=0$$

Divide 0 by v.

$$d=0$$

Solve for v (complex solution)

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

Solve for v

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