Possible Expression for a Function With Given Absolute Gradient

\[\mathbf{r} f(r)\]
  is sinusoidal and
\[|\mathbf{\nabla} f(r)|_{(-1,1,0)} =5\]
We want to find a possible expression for
\[\mathbf{r} f(r)\]
  is sinusoidal
\[\mathbf{\nabla} \cdot (\mathbf{r} f(r))=0\]
. Evaluating
\[\mathbf{\nabla} \cdot (\mathbf{r} f(r))\]
\[\begin{equation} \begin{aligned} \mathbf{\nabla} \cdot \mathbf{f} f(f) &= \frac{1}{r^2} \frac{\partial}{\partial r}(r^3 f(r)) \\ &= 3 f(f) + \frac{df}{dr} r=0 \end{aligned} \end{equation} \]

This is a differential equation for  
. We rearrange to give  
\[3 \frac{dr}{r} + \frac{df}{f}=0\]

Integrate to give
\[3 ln r +ln (f(r))=A\]

This can be rearranged to give
\[f(r)=\frac{e^A}{r^3} =\frac{B}{r^3}\]
\[|\mathbf{\nabla} (f(r)|_{(-1,1,0)}= |\frac{\partial f}{\partial r} |_{(-1,1,0)}= \frac{3B}{r^4}|_{(-1,1,0)}=5 \rightarrow \frac{3B}{4} =5 \rightarrow B= \frac{20}{3} \rightarrow f(r)=\frac{20}{3r^3}\]
  is a posible solution.