Solve 1/rd/dr(rdT(r)/dr)+frac{g_{0}}{k}=0

$$\frac { 1 } { r } \frac { d } { d r } ( r \frac { d T ( r ) } { d r } ) + \frac { g _ { 0 } } { k } = 0$$

Solve for g_0

$$g_{0}=-\frac{Tk}{r}$$

$$r\neq 0\text{ and }k\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by kr, the least common multiple of r,k.

$$k\times 1\frac{\mathrm{d}}{\mathrm{d}r}(r\frac{\mathrm{d}}{\mathrm{d}r}(Tr))+rg_{0}=0$$

Subtract k\times 1\frac{\mathrm{d}}{\mathrm{d}r}(r\frac{\mathrm{d}}{\mathrm{d}r}(Tr)) from both sides. Anything subtracted from zero gives its negation.

$$rg_{0}=-k\times 1\frac{\mathrm{d}}{\mathrm{d}r}(r\frac{\mathrm{d}}{\mathrm{d}r}(Tr))$$

Reorder the terms.

$$g_{0}r=-k\frac{\mathrm{d}}{\mathrm{d}r}(r\frac{\mathrm{d}}{\mathrm{d}r}(Tr))$$

The equation is in standard form.

$$rg_{0}=-Tk$$

Divide both sides by r.

$$\frac{rg_{0}}{r}=-\frac{Tk}{r}$$

Dividing by r undoes the multiplication by r.

$$g_{0}=-\frac{Tk}{r}$$

Solve for T

$$T=-\frac{g_{0}r}{k}$$

$$r\neq 0\text{ and }k\neq 0$$