where is a given function. This equation arises in electrostatics, elasticity theory and elsewhere.
(two dimensional form)
which arises in wave theory.
which arises in quantum mechanics. ( is Planck’s constant)
Transverse vibrations equation
for a homogeneous rod, where is the displacement at time of the cross section through .
All the PDEs we have discussed are second order (because the highest order derivatives that arise are second order) apart from the last example which is fourth order .