Solve -Gamma-rpartial*u^r=r_{e}u

$$- \Gamma - r \partial \cdot u ^ { r } = r _ { e } u$$

Solve for r_e

$$\left\{\begin{matrix}r_{e}=-\frac{r∂u^{r}+\Gamma }{u}\text{, }&u\neq 0\\r_{e}\in \mathrm{C}\text{, }&\Gamma =0\text{ and }u=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$r_{e}u=-\Gamma -r∂u^{r}$$

Reorder the terms.

$$r_{e}u=-r∂u^{r}-\Gamma$$

The equation is in standard form.

$$ur_{e}=-r∂u^{r}-\Gamma$$

Divide both sides by u.

$$\frac{ur_{e}}{u}=\frac{-r∂u^{r}-\Gamma }{u}$$

Dividing by u undoes the multiplication by u.

$$r_{e}=\frac{-r∂u^{r}-\Gamma }{u}$$

Divide -\Gamma -r∂u^{r} by u.

$$r_{e}=-\frac{r∂u^{r}+\Gamma }{u}$$