Completely classify the following differential equations $($complete the table$)$:

$(10$pts$)$

$\backslash$table$[[$#$,$PDE$/$System$,$Order,$\backslash$table$[[],[$Homogeneity$]],\backslash$table$[[$Dependent$],[$variables$]],\backslash$table$[[$Intendent$],[$variables$]],\backslash$table$[[$Can you guess its name? and is$],[$it generally solvable?$]]],[$a$,,,,,,],[$b$,,,,,,],[$c$,,,,,,],[$d$,,,,,,],[$e$,,,,,,]]$

a$)$ vec$(grad)*vec(grad)u=g(x,y,z)$ where $g$ is none zero $f(x,y,z)$

b$){}_t+u_x+v_y+w_z=g(t,x,y,z)$ where $$ and $u,v,w,$ and $g$ are none zero $f(t,x,y,z)$

c$){}_t+(u{)}_x+(v{)}_y+(w{)}_z=0$

$(u{)}_t+u(u{)}_x+v(u{)}_y+w(u{)}_z+p_x=(u_{}+u_{yy}+u_{zz})+f(x,y,z)$

$(v{)}_t+u(v{)}_x+v(v{)}_y+w(v{)}_z+p_y=(v_{}+v_{yy}+v_{zz})$

$(w{)}_t+u(w{)}_x+v(w{)}_y+w(w{)}_z+p_x=(w_{}+w_{yy}+w_{zz})$ for constant $,$ viscosity.

d$)$ Dynamic mass$/$m $($L: inductor$),$ damper$/$c $($R: resistance$),$ spring $(\frac{1}{C}$ : Capacitance$)$ system: $m.t^2{x}^{\text{'}\text{'}}(t)+c.t{x}^{\text{'}}(t)+k.x(t)=f(t)$ where $m,c,k$ are system constants, i$.$e$.$ mass, damping and stiffness.

e$)v_t+v{v}_x+v_x=g(t,x)$ where $g$ is none zero $f(t,x)$