Is the Integrator Linear in t?

[SuperCat]

Homework Statement

Determine whether the system is linear

Homework Equations

Superposition

The Attempt at a Solution

I am comfortable solving the case where the bounds are from negative infinity to t. I have provided an example of that solution I found online. I attempt to solve that problem in a similar fashion and conclude the system is linear, but according to the solutions (no work just answers) I was provided with the system is not linear.
I would appreciate if someone could tell me where I went wrong (why the same procedure can't be applied) or whether I am correct.

 

[phyzguy]

When you are asked whether a function x(t) is linear, the question is whether it is linear in t. In other words, is it true that x(a t1 + b t2) = a x(t1) + b x(t2)? You have shown that your function y(t) is linear in x, which it clearly is, but I doubt that is the question that was being asked. Perhaps the question should have been more explicit.

 

[SuperCat]

When you are asked whether a function x(t) is linear, the question is whether it is linear in t. In other words, is it true that x(a t1 + b t2) = a x(t1) + b x(t2)? You have shown that your function y(t) is linear in x, which it clearly is, but I doubt that is the question that was being asked. Perhaps the question should have been more explicit.

Thank you for your reply. I understand what you are saying. Most of the sample problems in my textbook demonstrate linearity by saying y(t) is linear in x.

Just to clarify, are you saying that the problem I posted is in-fact linear?

 

[phyzguy]

Just to clarify, are you saying that the problem I posted is in-fact linear?

No. To repeat, when you are asked whether a function x(t) is linear, the question is whether it is linear in t. In other words, is it true that x(a t1 + b t2) = a x(t1) + b x(t2)?

 

[SuperCat]

No. To repeat, when you are asked whether a function x(t) is linear, the question is whether it is linear in t. In other words, is it true that x(a t1 + b t2) = a x(t1) + b x(t2)?

Would that also suggest that the example problem I provided in 3 is non-linear?

 

[phyzguy]

Would that also suggest that the example problem I provided in 3 is non-linear?

Sorry, I don't understand your question. What exactly are you asking?

 

[SuperCat]

Sorry, I don't understand your question. What exactly are you asking?

The last image I posted where it says testing for linearity, that is testing just an integrator for linearity. I am aware that an integrator is linear. I want to know if that proof demonstrates it is linear in t.