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.

Thus ∆y_{(n+ + 1)} + Y_(n) = 2……………..(1)

and ∆y_{(n + 1)} + ∆² y_{(n − 1)}  = 1……………(2)

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

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

3. Derive the difference equation from Y_n = (A + Bn) (-3)ⁿ.

4. Derive the difference equation from _un = A2ⁿ + Bn

5. Derive the difference equation from

Y_n = (A + Bn) 2ⁿ

6. Write the difference equation ∆³y + ∆²yx + ∆y_x + y = 0 in the subscript notation :

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

Solution :

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

Solution: G

9. Form the difference equation generated by y=a2^x+b3^x+ c.

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

11. Form the difference equation from u̟_n = a 2ⁿ⁺¹