This is in fact the discrete analogue to a differential equation, called a difference equation.

Begin by searching for solutions to the associated homogeneous equation:

Set and substitute to obtain the quadratic , with solutions

.

The general solution to the homogeneous equation becomes , where the are constants.

Now we seek for a particular solution to the original equation. Since the right hand side is a quadratic polynomial in , we can try where are to be determined by substitution.

After this is done, the general solution to the original equation is .