A triple integral inside a PDE

Moving to spherical polar coordinates $\beta(m,n) \; = \; \int_0^\infty \int_0^{\frac12\pi} \int_0^{\frac12\pi} \frac{r^2 \sin\theta}{m + nr^4}\,dr\,d\theta\,d\phi \; = \; \tfrac12\pi \int_0^\infty \frac{r^2}{m + nr^4}\,dr \; = \; \frac{\pi}{2m^{\frac14}n^{\frac34}}\int_0^\infty \frac{s^2}{1 + s^4}\,ds$ and a simple substitution gives us that $\int_0^\infty \frac{s^2}{1 + s^4}\,ds \; = \; \tfrac14B(\tfrac14,\tfrac34) \; = \; \tfrac{\pi}{2\sqrt{2}}$ and hence $\beta(m,n) \; = \; \frac{\pi^2}{4\sqrt{2}m^{\frac14}n^{\frac34}}$ and hence $\frac{\partial^2}{\partial m \partial n}\beta(m,n) \; = \; \frac{3}{16mn}\beta(m,n)$ which makes $k =\boxed{3}$ .