Solve partialc=frac{varepsilon_{0}piD^2}{4d}2c

$$\partial c = \frac { \varepsilon _ { 0 } \pi D ^ { 2 } } { 4 d } 2 c$$

Solve for c (complex solution)

$$\left\{\begin{matrix}c=0\text{, }&d\neq 0\\c\in \mathrm{C}\text{, }&∂=\frac{\pi \epsilon _{0}D^{2}}{2d}\text{ and }d\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 4d.

$$∂c\times 4d=\epsilon _{0}\pi D^{2}\times 2c$$

Subtract \epsilon _{0}\pi D^{2}\times 2c from both sides.

$$∂c\times 4d-\epsilon _{0}\pi D^{2}\times 2c=0$$

Multiply -1 and 2 to get -2.

$$∂c\times 4d-2\epsilon _{0}\pi D^{2}c=0$$

Combine all terms containing c.

$$\left(∂\times 4d-2\epsilon _{0}\pi D^{2}\right)c=0$$

The equation is in standard form.

$$\left(4d∂-2\pi \epsilon _{0}D^{2}\right)c=0$$

Divide 0 by 4∂d-2\epsilon _{0}\pi D^{2}.

$$c=0$$

Solve for c

$$\left\{\begin{matrix}c=0\text{, }&d\neq 0\\c\in \mathrm{R}\text{, }&∂=\frac{\pi \epsilon _{0}D^{2}}{2d}\text{ and }d\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 4d.

$$∂c\times 4d=\epsilon _{0}\pi D^{2}\times 2c$$

Subtract \epsilon _{0}\pi D^{2}\times 2c from both sides.

$$∂c\times 4d-\epsilon _{0}\pi D^{2}\times 2c=0$$

Multiply -1 and 2 to get -2.

$$∂c\times 4d-2\epsilon _{0}\pi D^{2}c=0$$

Combine all terms containing c.

$$\left(∂\times 4d-2\epsilon _{0}\pi D^{2}\right)c=0$$

The equation is in standard form.

$$\left(4d∂-2\pi \epsilon _{0}D^{2}\right)c=0$$

Divide 0 by 4∂d-2\epsilon _{0}\pi D^{2}.

$$c=0$$

Solve for D (complex solution)

$$\left\{\begin{matrix}D=-\left(\pi \epsilon _{0}\right)^{-\frac{1}{2}}\sqrt{d}\sqrt{2∂}\text{; }D=\left(\pi \epsilon _{0}\right)^{-\frac{1}{2}}\sqrt{d}\sqrt{2∂}\text{, }&\epsilon _{0}\neq 0\text{ and }d\neq 0\\D\in \mathrm{C}\text{, }&\left(c=0\text{ and }d\neq 0\right)\text{ or }\left(∂=0\text{ and }\epsilon _{0}=0\text{ and }d\neq 0\right)\end{matrix}\right.$$

Solve for D

$$\left\{\begin{matrix}D=\sqrt{\frac{2d∂}{\pi \epsilon _{0}}}\text{; }D=-\sqrt{\frac{2d∂}{\pi \epsilon _{0}}}\text{, }&\left(∂\geq 0\text{ and }\epsilon _{0}<0\text{ and }d<0\right)\text{ or }\left(∂\geq 0\text{ and }\epsilon _{0}>0\text{ and }d>0\right)\text{ or }\left(∂\leq 0\text{ and }\epsilon _{0}>0\text{ and }d<0\right)\text{ or }\left(∂\leq 0\text{ and }\epsilon _{0}<0\text{ and }d>0\right)\\D\in \mathrm{R}\text{, }&\left(c=0\text{ and }d\neq 0\right)\text{ or }\left(∂=0\text{ and }\epsilon _{0}=0\text{ and }d\neq 0\right)\end{matrix}\right.$$