Solve I=rho_{3}sqrt{2m+1}dx

$$I = \rho _ { 3 } \sqrt { 2 m + 1 } d x$$

Solve for d (complex solution)

$$\left\{\begin{matrix}d=\frac{\left(2m+1\right)^{-\frac{1}{2}}I}{x\rho _{3}}\text{, }&x\neq 0\text{ and }m\neq -\frac{1}{2}\text{ and }\rho _{3}\neq 0\\d\in \mathrm{C}\text{, }&\left(x=0\text{ or }m=-\frac{1}{2}\text{ or }\rho _{3}=0\right)\text{ and }I=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$\rho _{3}\sqrt{2m+1}dx=I$$

The equation is in standard form.

$$\sqrt{2m+1}x\rho _{3}d=I$$

Divide both sides by \rho _{3}\sqrt{2m+1}x.

$$\frac{\sqrt{2m+1}x\rho _{3}d}{\sqrt{2m+1}x\rho _{3}}=\frac{I}{\sqrt{2m+1}x\rho _{3}}$$

Dividing by \rho _{3}\sqrt{2m+1}x undoes the multiplication by \rho _{3}\sqrt{2m+1}x.

$$d=\frac{I}{\sqrt{2m+1}x\rho _{3}}$$

Divide I by \rho _{3}\sqrt{2m+1}x.

$$d=\frac{\left(2m+1\right)^{-\frac{1}{2}}I}{x\rho _{3}}$$

Solve for d

$$\left\{\begin{matrix}d=\frac{I}{\sqrt{2m+1}x\rho _{3}}\text{, }&x\neq 0\text{ and }\rho _{3}\neq 0\text{ and }m>-\frac{1}{2}\\d\in \mathrm{R}\text{, }&\left(I=0\text{ and }\rho _{3}=0\text{ and }x\neq 0\text{ and }m>-\frac{1}{2}\right)\text{ or }\left(I=0\text{ and }m=-\frac{1}{2}\right)\text{ or }\left(I=0\text{ and }m\geq -\frac{1}{2}\text{ and }x=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$\rho _{3}\sqrt{2m+1}dx=I$$

The equation is in standard form.

$$\sqrt{2m+1}x\rho _{3}d=I$$

Divide both sides by \rho _{3}\sqrt{2m+1}x.

$$\frac{\sqrt{2m+1}x\rho _{3}d}{\sqrt{2m+1}x\rho _{3}}=\frac{I}{\sqrt{2m+1}x\rho _{3}}$$

Dividing by \rho _{3}\sqrt{2m+1}x undoes the multiplication by \rho _{3}\sqrt{2m+1}x.

$$d=\frac{I}{\sqrt{2m+1}x\rho _{3}}$$

Solve for I (complex solution)

$$I=\sqrt{2m+1}dx\rho _{3}$$

Solve for I

$$I=\sqrt{2m+1}dx\rho _{3}$$

$$m\geq -\frac{1}{2}$$