Problem: Let $\mathbf{r} = x \hat{i} + y \hat{j} + z \hat{k}$ be the position vector of a general point in $3$-space, and let $s= |\mathbf{r}|$ be the length of $\mathbf{r}$. For what value of the constant $k$ is the vectorfield $s^k \mathbf{r}$ solenoidal except at the origin? Find all functions $f(s)$, differentiable for $s >0$, such that $f(s)\mathbf{r}$ is solenoidal everywhere except at the origin in $3$-space.
Attempt at solution: We demand dat $\nabla \cdot (s^k \mathbf{r}) = 0$. Making use of a vector identity, this equals \begin{align*} (\nabla s^k) \cdot \mathbf{r} + s^k ( \nabla \cdot \mathbf{r}) = 0. \end{align*} For the first term on the left, we have \begin{align*} \nabla s^k &= \frac{1}{2}k (x^2 + y^2 + z^2)^{\frac{1}{2}k - 1} 2x \hat{i} + \frac{1}{2}k (x^2 + y^2 + z^2)^{\frac{1}{2}k -1} 2y \hat{j} + \frac{1}{2}k (...)^{\frac{1}{2}k -1} 2z \hat{k} \\ &= xk(...)^{\frac{1}{2}k - 1} \hat{i} + yk(...)^{\frac{1}{2}k-1} \hat{j} + zk(...)^{\frac{1}{2}k-1} \hat{k} \end{align*} Then \begin{align*} (\nabla s^k) \cdot \mathbf{r} = x^2k(x^2 + y^2 + z^2)^{\frac{1}{2}k-1} + y^2k(...)^{\frac{1}{2}k-1} + z^2 k (...)^{\frac{1}{2}k-1} \end{align*}
For the term on the right we have \begin{align*} s^k ( \nabla \cdot \mathbf{r}) = 3s^k = 3(x^2 + y^2 + z^2)^{\frac{1}{2}k} \end{align*} Putting everything together, we demand that \begin{align*} x^2k(x^2 + y^2 + z^2)^{\frac{1}{2}k-1} + y^2k(...)^{\frac{1}{2}k-1} + z^2 k (...)^{\frac{1}{2}k-1} + 3(x^2 + y^2 + z^2)^{\frac{1}{2}k} &= 0, \end{align*} or (collecting terms) \begin{align*} k(x^2 + y^2 + z^2)^{\frac{1}{2}k-1} s^2 + 3(x^2 + y^2 + z^2)^{\frac{1}{2}k} = 0. \end{align*} This is equivalent with \begin{align*} k(s^2)^{\frac{1}{2}k-1} s^2 + 3(s^2)^{\frac{1}{2}k} = 0 \end{align*} Then I'm not sure how to rewrite this to find the $k$ needed. Any help please? Also, about the last question, how should I find all the $f(s)'s$?
$$\nabla f(s)=\frac{df}{ds}\left(\frac{\partial s}{\partial x},\frac{\partial s}{\partial y},\frac{\partial s}{\partial z}\right)\\ =\frac{df}{ds}\left(\frac xs,\frac ys,\frac zs\right)\\ \nabla f(s).r=f'(s)\left(\frac{x}{s},\frac{y}{s},\frac{z}{s}\right)\cdot(x,y,z)\\ =sf'(s)+3f(s)=0$$