Consider functions of real variables $u(x,y,z,x',y',z')$,$\quad$$v(x,y,z,x',y',z')$,$\quad$$w(x,y,z,x',y',z')$,$\quad$ $t(x,y,z,x',y',z')$,$\quad$$h(x,y,z,x',y',z')$,$\quad$$g(x,y,z,x',y',z')$,$\quad$$f(x,y,z,x',y',z')=e^{-(x^2+y^2+z^2)}e^{-(x'^2+y'^2+z'^2)}$
$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}(u^2+v^2+w^2+t^2+h^2+g^2)dxdydzdx'dy'dz' = 1$
$\frac {\partial u}{\partial x}+\frac {\partial v}{\partial y}+\frac {\partial w}{\partial z}+\frac {\partial t}{\partial x'}+\frac {\partial h}{\partial y'}+\frac {\partial g}{\partial z'}=f$
written short-hand as $\nabla .(u+v+w)+\nabla ' .(t+h+g)=f$
Please how do I solve the PDE above?
based on the solution of Poisson equation $\nabla .F= f$ :
$u=a e^{-(x'^2+y'^2+z'^2)}\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-(x'^2+y'^2+z'^2)}\frac {x'-x}{4\pi r'^3}dx'dy'dz'$
$v=be^{-(x'^2+y'^2+z'^2)}\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-(x'^2+y'^2+z'^2)}\frac {y'-y}{4\pi r'^3}dx'dy'dz'$
$w=ce^{-(x'^2+y'^2+z'^2)}\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-(x'^2+y'^2+z'^2)}\frac {z'-z}{4\pi r'^3}dx'dy'dz'$
similar expressions works for $t,h,g$
constants $a,b,c$ can be dterminined from the constraint.