Viete theorem, named after French mathematician Franciscus Viete relates the coefficients of a polynomial equation to sums and products of its roots. It states:
For quadratic equation , with roots ,
This is easy to prove. We know that the roots of are
In fact, the converse is also true. If two given numbers are such that
are the roots of .
This is also easy to prove. From (2) we have. Hence, (2) implies that , or
Since are symmetric in both (1) and (2), (3) implies that is also the root of .
Let us consider the second-order linear ordinary differential equation with constant coefficients:
Let be the roots of quadratic equation with unknown .
By Viete’s theorem,
Therefore, (4) can be written as
Rearrange the terms, we have
(5), a second-order equation is reduced to a first-order equation
To obtain , we solve two first-order equations, (7) for first, then (6) for .
We are now ready to show that any solution obtained as described above is also a solution of (1):
Let be the result of solving (7) for then (6) for ,
(8) tells that is a solution of (5).
The fact that (5) is equivalent to (1) implies , a solution of (5) is also a solution of (1)