Is this a linear system?

Since the integral is a linear operator, it can be pulled out of the integral and you get (Tx_1)(t) = k(x(t) + \int (t - \tau) x(\tau) \,d\tau) = k(Tx)(t).In summary, the conversation discusses the linearity of a system, with the given equation y(t) = x(t) + \int (t - \tau)x(\tau)d\tau. It is concluded that the system is not linear, as shown by the fact that T[kx(t)] does not equal kT[x(t)]. The reason for this is because x(\tau) should also be multiplied by k inside the integral. This is shown through
  • #1
360
0
y(t) = x(t) + [tex]\int[/tex] (t - [tex]\tau[/tex])x([tex]\tau[/tex])d[tex]\tau[/tex]

for it to be linear, T[kx(t)] = kT[x(t)] so i have
T[kx(t)] = kx(t) + [tex]\int[/tex] (t - [tex]\tau[/tex])x([tex]\tau[/tex])d[tex]\tau[/tex]
and
kT[x(t)] = k[x(t) + [tex]\int[/tex] (t - [tex]\tau[/tex])x([tex]\tau[/tex])d[tex]\tau[/tex]] = kx(t) + k[tex]\int[/tex] (t - [tex]\tau[/tex])x([tex]\tau[/tex])d[tex]\tau[/tex]
so they aren't equal and aren't linear. however, I'm not sure about this answer because for the first part, T[kx(t)], I'm not sure if x([tex]\tau[/tex]) should also be multiplied by k, making it a linear system

any help would be appreciated. thx.
 
Physics news on Phys.org
  • #2
You should, in fact, be multiplying [tex]x(\tau)[/tex] by [tex]k[/tex] inside the integral -- simple substitution is how you see this. If [tex](Tx)(t) = x(t) + \int (t - \tau) x(\tau) \,d\tau[/tex] and [tex]x_1(t) = kx(t)[/tex] then [tex](Tx_1)(t) = x_1(t) + \int (t - \tau) x_1(\tau) \,d\tau = kx(t) + \int (t - \tau) kx(\tau) \,d\tau[/tex].
 

1. What is a linear system?

A linear system is a mathematical model that consists of a set of linear equations, where the variables are raised to the first power and there are no products or powers of variables. These equations can be solved using various methods, such as substitution or elimination, to find the values of the variables.

2. How do I know if a system is linear?

A system is linear if all of the equations in the system are linear equations. This means that each equation must have the variables raised to the first power and there should be no products or powers of variables. If any equation in the system does not meet these criteria, then the system is not linear.

3. Can a system be both linear and non-linear?

No, a system cannot be both linear and non-linear. A system is either linear or non-linear, depending on the type of equations it contains. If even one equation in the system is non-linear, then the entire system is considered non-linear.

4. What are some examples of linear and non-linear systems?

Examples of linear systems include systems of linear equations, such as 2x + 3y = 10 and 5x - 2y = 8. Non-linear systems can include equations with variables raised to powers other than 1, such as x^2 + 3y = 7 and 2x + y^3 = 5. Additionally, systems with trigonometric or exponential functions are also considered non-linear.

5. Why is it important to determine if a system is linear?

Determining if a system is linear is important because it helps us understand the behavior and properties of the system. Linear systems have unique characteristics, such as having a constant slope, and can be solved using efficient methods. On the other hand, non-linear systems can have more complex behaviors and may require more advanced techniques to solve.

Suggested for: Is this a linear system?

Replies
2
Views
410
Replies
8
Views
700
Replies
7
Views
750
Replies
13
Views
1K
Replies
5
Views
861
Replies
4
Views
1K
Replies
16
Views
1K
Back
Top