## Linear Recurrence Relations with Constant CoefficientsA Recurrence Relations is called linear if its degree is one. The general form of linear recurrence relation with constant coefficient is C Where C A solution of a recurrence relation in any function which satisfies the given equation. ## Linear Homogeneous Recurrence Relations with Constant Coefficients:The equation is said to be linear homogeneous difference equation if and only if R (n) = 0 and it will be of order n. The equation is said to be linear non-homogeneous difference equation if R (n) ≠ 0.
A linear homogeneous difference equation with constant coefficients is given by C Where The solution of the equation (i) is of the form , where ∝ Substitute the values of A ∝ C After simplifying equation (ii), we have C The equation (iii) is called the characteristics equation of the difference equation. If ∝ To find the solution of the linear homogeneous difference equations, we have the four cases that are discussed as follows:
Thus, are all solutions of equation (i). Also, we have are all solutions of equation (i). The sums of solutions are also solutions. Hence, the homogeneous solutions of the difference equation are
If ∝ If ∝ Similarly, if root ∝ (A The solution to the homogeneous equation.
If α+iβ is the root of the characteristics equation, then α-iβ is also the root, where α and β are real. Thus, (α+iβ) (α+iβ) Is also a solution to the characteristics equation, where A
When the characteristics equation has repeated imaginary roots, (C Is the solution to the homogeneous equation.
^{2}-3s+2=0 or (s-1)(s-2)=0⇒ s = 1, 2 Therefore, the homogeneous solution of the equation is given by a_{r}=C^{1}_{r}+C^{2}.2^{r}.
^{2}-6s+1=0 or (3s-1)^{2}=0⇒ s = and Therefore, the homogeneous solution of the equation is given by
Therefore, the homogeneous solution of the equation is
Therefore, the homogeneous solution of the equation is given by y Next TopicParticular Solution |