Show that Txyzt ekt cosx cosy cosz satisfies the threedim

Show that T(x,y,z,t) = e-kt (cosx + cosy + cosz) satisfies the three-dimensional heat equation: k(delta2T/delta x2 + delta2 T/delta y2 + delta2 T/delta z2) = delta T/delta t

Solution

LHS = k( d²T/dx² +d²T/dy² + d²T/d²z)
=k(d²(e^-kt (cosx+cosy+cosz)/dx² +d² (e^-kt (cosx+cosy +cosz)/dy² +d² (e^-kt (cosx+cosy +cosz)/dz² )
=k(-e^-kt cosx -e^-kt cosy- e^-kt cosz)

=-k(e^-kt( cosx +cosy +cosz)

RHS= dT/dt = d(e^-kt (cosx+cosy+cosz))/dt = -ke^-kt (cosx+cosy+cosz)

LHS=RHS

so given equation satisfy three dimensional heat equation