How to find the equation of line in spherical co-ordinate system? I am trying to find the equation of the curve between two points in spherical co-ordinate whose length is shortest, i.e. find the equation of line in spherical co-ordinate system. Here is my work so far, ![enter image description here]( Did I do some mistake somewhere? Because I am stuck and don't know how to solve the differential equation, $$\frac{r}{\sqrt{r'^2+r^2}}-\frac{d}{d\theta}\frac{r'}{\sqrt{r'^2+r^2}} = 0$$ Please tell me how to proceed. P.S. I made a mistake in the final equation, forgot the square-root and r prime. Please ignore.

Why not simply $$ ax+by+c=0 \implies ar\cos\theta+br\sin\theta+c=0\implies r=-\frac{c}{a\cos\theta+b\sin\theta} $$

Anyway, you should perform the derivative, obtaining $$ \frac{r}{\sqrt{r'^2+r^2}}-\frac{r''}{\sqrt{r'^2+r^2}}+\frac{r'(2r'r''+2rr')}{2\sqrt{(r'^2+r^2)^3}} = 0 $$ and further simplify to $$ -\frac{r(rr''-2r'^2-r^2)}{\sqrt{(r'^2+r^2)^3}} = 0 $$