# physics – Solving two coupled partial differential equations

I am trying to solve the following system of two coupled partial differential equations (both equations equal 0):

Here, $$V$$ and $$Y$$ are functions that only depend on $$r$$, i.e., $$V=V(r)$$ and $$Y=Y(r)$$. $$alpha$$, $$beta$$, $$gamma$$ and $$k^{-2}$$ are just constants, $$H^{Lmu}_mu=0$$ and $$H^{Li}_i-H^{Lt}_t=0$$. I’ve tried to use DSolve but it doesn’t work:

``````DSolve({2*(3*b - a))*1/r^4*(r^2*v'(r))'' - ck*1/r^2*(r^2*v'(r))' - 4*(3*b - a)*1/r^2*(r^2*Y'(r))' + 2*ckY(r) == 0, 2*b*1/r^4*(r^2*v'(r))'' - ck*1/r^2*(r^2*v'(r))' + 2*(a - 2*b)*1/r^2*(r^2*Y'(r))' == 0}, {v, Y}, r)
``````

I don’t know much about mathematica and this problem is untractable without numerical methods. Can someone send some help? Thank you so much.