u_x+u_y+u_z+u=0

u(x,y,z)=\varphi(x)\psi(y)\omega(z)

\varphi'(x)\psi(y)\omega(z)+\varphi(x)\psi'(y)\ome  ga(z)+\varphi(x)\psi(y)\omega'(z)+\varphi(x)\psi(y  )\omega(z)=0

\varphi'(x)\psi(y)\omega(z)+\varphi(x)[\psi'(y)\omega(z)+\psi(y)\omega'(z)+\psi(y)\omega(  z)]=0

\displaystyle\frac{\varphi'(x)}{\varphi(x)}=-\left[\frac{\psi'(y)\omega(z)+\psi(y)\omega'(z)+\psi(y)\  omega(z)}{\psi(y)\omega(z)}\right]

\varphi'(x)-\lambda\varphi(x)=0\Rightarrow m=\lambda

\varphi(x)=\exp{(x\lambda)}

\psi'(y)\omega(z)+\psi(y)\omega'(z)+\psi(y)\omega(  z)+\lambda\psi(y)\omega(z)=0

\psi'(y)\omega(z)+\psi(y)[\omega'(z)+\omega(z)+\lambda\omega(z)]=0

\displaystyle\frac{\psi'(y)}{\psi(y)}=-\left[\frac{\omega'(z)+\omega(z)+\lambda\omega(z)}{\omeg  a(z)}\right]

\psi'(y)-\mu\psi(y)=0\Rightarrow n=\mu

\psi(y)=\exp{(y\mu)}

\omega'(z)+\omega(z)+\lambda\omega(z)+\mu\omega(z)  =0

t+1+\lambda+\mu=0\Rightarrow t=-(1+\lambda+\mu)

\omega(z)=\exp{-z(1+\lambda+\mu)}

u(x,y,z)=C\exp{[x\lambda+y\mu-z(1+\lambda+\mu)]}

I was original going to ask why this didn't work when I checked it but now it worked. However, since I typed it all out, I am going to post it for others to look at.