Formation of difference equations

Formation of difference equations :

Def. Difference equations: A difference equation is a relation between the differences of an unknown function at one or more general values of the argument.

are difference equations.

Def: Order of a difference equation :

The order of a difference equation is the difference between the largest and the smallest arguments occurring in the difference equation divided by the unit of increment.

Def: Solution of a difference equation :

The solution of a difference equation is an expression for y(n) which satisfies the given difference equation.

Def: The general solution of a difference equation: The general solution of a difference equation is that in which the number of arbitrary constants is equal to the order of the difference equation.

Def: The particular solution of a difference equation:

A particular solution is that the solution which is obtained from the general solution by giving particular values to the constants.

XII. Formation of difference equations :

1. Form the difference equation corresponding to the family of curves y = ax + bx2

Solution :

Eliminating a and b from (1), (2) and (3), we get

2. From yn = a2n + b(-2)n, derive a difference equation by eliminating the arbitrary constants.

Solution: 

Eliminating a (2)n & b (−2)n from (1), (2) and (3), we get

which is the desired difference equation.

3. Derive the difference equation from

yn = (A + Bn) (−3)n.

Solution :

Eliminating A(-3)n and B(-3)n from (1), (2) & (3), we get

4. Derive the difference equation from

un = A2n + Bn

Solution :

Given :

Eliminating A2n and B from (1), (2) and (3), we get

5. Derive the difference equation from

yn = (A + Bn) 2n

Solution:

Given :

Eliminating A 2n and B 2n from (1), (2) and (3), we get

Note: Formulae

6. Write the difference equation Δ3yx + Δ2 yx + Δ yx + yx = 0 in the subscript notation:

Solution: 

7. Find the difference equation satisfied by y = ax2 – bx.

Solution :

Eliminating a and b from (1), (2) and (3), we get

8. Form the difference equation generated by yx = ax + b 2x

Solution: 

Eliminating a and b 2x from (1), (2) and (3), we get

9. Form the difference equation generated by yx = a2x + b3x + c.

Solution: 

Eliminating a 2x, b 3x, c from (1), (2), (3) and (4), we get

10. Form the difference equation from yn = a + b3n.

Solution:

Eliminating a and b 3n from (1), (2) and (3) we get

11. Form the difference equation from un = a2n+1

Solution :

Eliminating a 2n+1 from (1) & (2) we get