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# Solution of Differential Equation

One sided Z transform is very efficient tool for the solution of difference equations with nonzero initial condition. System function of LSI system can be obtained from its difference equation.

SOLUTION OF DIFFERENTIAL EQUATION

One sided Z transform is very efficient tool for the solution of difference equations with nonzero initial condition. System function of LSI system can be obtained from its difference equation.

1.  Difference equations are used to find out the relation between input and output sequences. It is also used to relate system function H(z) and Z transform.

2.     The transfer function H(ω) can be obtained from system function H(z) by putting z=ejω.

Magnitude and phase response plot can be obtained by putting various values of ω.

Tutorial problems:

Q) A difference equation of the system is given below

Y(n)= 0.5 y(n-1) + x(n)

Determine

a) System function

b) Pole zero plot

c) Unit sample response

Q) A difference equation of the system is given below

Y(n)= 0.7 y(n-1) – 0.12 y(n-2) + x(n-1) + x(n-2)

a) System Function

b) Pole zero plot

c) Response of system to the input x(n) = nu(n)

d) Is the system stable? Comment on the result.

Q) A difference equation of the system is given below

Y(n)= 0.5 x(n) + 0.5 x(n-1)

Determine a) System function

b) Pole zero plot

c) Unit sample response

d) Transfer function

e) Magnitude and phase plot

Q) A difference equation of the system is given below

a. Y(n)= 0.5 y(n-1) + x(n) + x(n-1)

b. Y(n)= x(n) + 3x(n-1) + 3x(n-2) + x(n-3)

a) System Function

b) Pole zero plot

c) Unit sample response

d) Find values of y(n) for n=0,1,2,3,4,5 for x(n)= δ(n) for no initial condition.

Q) Solve second order difference equation

2x(n-2) – 3x(n-1) + x(n) = 3n-2 with x(-2)=-4/9 and x(-1)=-1/3.

Q) Solve second order difference equation

x(n+2) + 3x(n+1) + 2x(n) with x(0)=0 and x(1)=1.

Q) Find the response of the system by using Z transform

x(n+2) - 5x(n+1) + 6x(n)= u(n) with x(0)=0 and x(1)=1.

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