The last equation is a homogeneous linear second order differential equation with constant coefficients:
where
For these simple equations, we look for a solution of the form
Substituting these values we get
Dividing through by 
gives
which is the characteristic equation of the differential equation. The solution of this equation can be derived from the quadratic equation
Substituting the values gives
The two solutions to the differential equation are
And the complete solution is
Recalling that
