andrew said:Hi all,
Could someone please explain to me the process involved in converting an inhomogeneous recurrence to a homogeneous recurrence, I'm completely confused as to how it works.Thanks
chisigma said:For semplicity we suppose that we have linear second order difference equations. A homogeneous difference equation is written as...
$\displaystyle a_{n+2} + c_{1}\ a_{n+1}+ c_{0}\ a_{n} = 0\ (1)$
An inhomogeneous difference equation is written as...
$\displaystyle a_{n+2} + c_{1}\ a_{n+1} + c_{0}\ a_{n} = b_{n}\ (2)$
Kind regards
$\chi$ $\sigma$
andrew said:Could you possibly provide an example, this would help me understand it a bit better.
chisigma said:An example of linear homogeneous second order difference equation is here...
http://mathhelpboards.com/discrete-mathematics-set-theory-logic-15/recursive-sequences-finding-their-expressions-10478.html#post48615
Kind regards
$\chi$ $\sigma$
andrew said:Sorry, I meant an example of an inhomogeneous recurrence relation, I understand how to solve a homogeneous recurrence relation, but converting an inhomogeneous recurrence is where I am struggling.
An inhomogeneous recurrence relation is a mathematical equation that describes the relationship between successive terms in a sequence, where the terms on the right side of the equation are not all constants. This means that the equation contains terms that are dependent on previous terms in the sequence, as well as terms that are not related to the previous terms.
A homogeneous recurrence relation only contains terms that are dependent on previous terms in the sequence, while an inhomogeneous recurrence relation contains additional terms that are not related to the previous terms. Inhomogeneous recurrence relations are often more complex and difficult to solve than homogeneous ones.
Inhomogeneous recurrence relations can be found in various fields such as physics, economics, and biology. For example, in physics, the motion of a pendulum can be described by an inhomogeneous recurrence relation, where the acceleration term is dependent on the previous position and velocity of the pendulum, as well as external forces such as friction.
The general solution for an inhomogeneous recurrence relation is a combination of the particular solution and the complementary solution. The particular solution is a specific solution to the inhomogeneous equation, while the complementary solution is a solution to the corresponding homogeneous equation. The particular solution can be found using methods such as variation of parameters or undetermined coefficients.
Inhomogeneous recurrence relations are widely used in various fields of science and engineering to model and solve problems related to sequences and systems. They can be used to describe complex phenomena, such as population growth, chemical reactions, and signal processing. Inhomogeneous recurrence relations also have practical applications in computing, where they are used to create efficient algorithms for tasks such as sorting and searching.