Solving Difference Equations using Z-transforms

This assignment focuses on solving difference equations using Z-transforms. It covers two questions, each with its own MATLAB code, which helps students understand how to apply Z-transforms to solve difference equations. The first question involves solving a non-homogeneous difference equation, while the second question deals with solving a homogeneous linear difference equation. The assignment provides a detailed understanding of the MATLAB code and helps students grasp the concept of Z-transforms in solving difference equations.

APPLICATIONS OF DIFFERENTIAL
AND DIFFERENCE EQUATIONS –
MAT2002
LAB ASSIGNMENT – 05
NAME: S. UDAY KIRAN
REGNO.: 18MIS0050
FACULTY NAME: ARUNA.K
SLOT: L47+L48
EXPERIMENT – 5A
Z-transforms and their applications for solving
Difference equations
QUESTION 1:
Solve yn+2 −4yn+1 + 3yn = n(2^n), n ≥ 0 y0 = 0, y1 = 0
using Z-transform. Plot the solution
AIM:
To find the solution of the difference equation
using the initial conditions given and Z-transform
MATLAB CODE:
clear all
clc
syms n z y(n) Y
A=input('Enter y(n+2),y(n+1),y(n) coeff in vector
form');

f=input('Enter the non-homogeneous part f(x): ');
cond=input('Enter the initial conditions in form
[y(0),y(1)]: ')
a=A(1);b=A(2);c=A(3);
eqn=ztrans(a*y(n+2)+b*y(n+1)+c*y(n)-f)
y0=cond(1);dy0=cond(2);
eqn=subs(eqn,
{str2sym
('ztrans(y(n),n,z)'),str2sym('y(0)'),str2sym('y(1)')}
,{Y,y0,dy0});
Y=solve(eqn,Y);
y=simplify(iztrans(Y,z,n))
m=0:10;
disp('The solution of the differential equation : ')
disp(y);
y=subs(y,n,m)
stem(y)
INPUT:
Enter y(n+2),y(n+1),y(n) coeff in vector form
[1 -4 3]
Enter the non-homogeneous part f(x):
2*(2^n)
Enter the initial conditions in form [y(0),y(1)]:
[0 0]
OUTPUT:
The solution of the differential equation :
3^n - 2*2^n + 1
y =
[ 0, 0, 2, 12, 50, 180, 602, 1932, 6050, 18660,
57002]

EXPERIMENT – 5B
Solution of homogeneous linear difference
equations
QUESTION 2:
Solve 2yn+2−7yn+1 +3yn = 0 subject to the given
conditions y0 = 1, y1 = 1.Plot the solution.
AIM:
To solve the given difference equation.
MATLAB CODE:
clc
clear all
syms c1 c2 n m
coeff=input('Enter y(n+2),y(n+1),y(n) coeff in vector
form');
n0=input('Enter initial values n0 and n1');
inic=input('Enter the initial conditions in form
[y(n0),y(n1)]: ');
m=roots([coeff(1) coeff(2) coeff(3)])
%or AE=coeff(1)*m^2+coeff(2)*m+coeff(3);m=solve(AE);
disc=coeff(2)^2-4*coeff(1)*coeff(3);
if disc>0
y1=m(1)^n;%first solution
y2=m(2)^(n);%second solution(independent)
elseif disc==0
y1=m(1)^n;
y2=n*m(1)^n;
else
a=real(m(1));%real part of m
b=abs(imag(m(1)));% imaginary part of m
r=sqrt(a^2+b^2);
theta=atan(abs(b)/abs(a));
y1=r^n*cos(theta*n);
y2=r^n*sin(theta*n);
end
y=c1*y1+c2*y2 %complementary solution

[c1 c2]=solve(subs(y,n,n0(1))-
inic(1),subs(y,n,n0(2))-inic(2))
disp('General solution: ');
yn=simplify(subs(y));
disp(yn);
n1=0:10;
stem(n1,subs(yn,n,n1))
INPUT:
Enter y(n+2),y(n+1),y(n) coeff in vector form
[2 -7 3]
Enter initial values n0 and n1
[0 1]
Enter the initial conditions in form [y(n0),y(n1)]:
[1 1]
OUTPUT:
General solution:
(6^n/5 + 4/5)/2^n