Testing Linearity & Time Invariance

Hello all,

So I have this small assignment for my systems and signals class where i need to take a group of equations and test them for linearity and time invariance. Well I have something thats working for the simple functions we have to test for but there are a few that im struggling to figure out. If anyone could point me in the right direction I'd greatly appreciate it. The functions to test are as follows:

1) y(t) = 2x(t)

2) y(t) = x(t) + 1

3) y(t) = ln(x(t))

4) y(t) = x(t)*x(t)

5) y'(t) + 2y(t) = 2x(t)

6) y(t) = x(t)*u(t)

7) y(t) = tu(t)

I am able to test 1-4, but 5-7 are the ones I'm struggling on

below is the code I got working for the simple functions.

function plotlti(T)
% LTI Check linearity and time invariance of systems
% T is a string which gives the transformation of the system
% i.e. T = '2*x' would indicate y(t)=2*x(t)
%
% Your Name Date
additivity(T);
scaling(T);
TI1(T);
TI2(T);
end

function additivity(T)
clf
t1 = -5:0.05:5; % fine time scale
t2 = -5:0.5:5; % coarse time scale
% Additivity Experiment #1
x1 = u(t1); % create two steps
x2 = u(t1);
x = x1 + x2; % add them
e1 = eval(T); % apply transformation
% Additivity Experiment #2
x = u(t2); % create a step
y1 = eval(T); % transform it to form y1
y2 = y1; % use the same output as y2
e2 = y1 + y2; % add them up
plot(t1,e1);
hold on;
plot(t2,e2);
end

function scaling(T)
clf
t1 = -5:0.05:5; % fine time scale
t2 = -5:0.5:5; % coarse time scale
% Scalability Experiment #1
x1 = 2*u(t1); % scale step function by a factor of 2
x = x1; % make x = to the scaled x(t)
e1 = eval(T); % apply transformation for the scaled x(t)
% Additivity Experiment #2
x = u(t2); % create a step
y1 = eval(T); % transform it to form y1
e2 = 2*y1; % make e2 our scaled y(t) by a factor of 2
plot(t1,e1);
hold on;
plot(t2,e2);
end 
 
 
function TI2(T)
clf
t1 = -5:0.05:5; % fine time scale
t2 = -5:0.5:5; % coarse time scale
% TI -1s Experiment #1
x= u(t1-1); % shift the step function to the right 1s
e1 = eval(T); % apply transformation for the shifted x(t)
% Additivity Experiment #2
x = u(t2); % create a step
y1 = eval(T); % transform it to form y1
e2 = y1; % make e2 our y(t)
plot(t1,e1);
hold on;
plot(t2+1,e2);
end


function TI1(T)
clf
t1 = -5:0.05:5; % fine time scale
t2 = -5:0.5:5; % coarse time scale
% TI -1s Experiment #1
x= u(t1+1); % shift the step function to the right 1s
e1 = eval(T); % apply transformation for the shifted x(t)
% Additivity Experiment #2
x = u(t2); % create a step
y1 = eval(T); % transform it to form y1
e2 = y1; % make e2 our y(t)
plot(t1,e1);
hold on;
plot(t2-1,e2);
end

Answer 1

[Baba Maligeli]

Y(n)=x^2(n)

[Baba Maligeli]

Answer it if possible to check time invariance