# Is the System y*y' + 3y = x Nonlinear?

• elimenohpee
In summary, the system described is nonlinear because the input and output do not have the same form. This can be seen by multiplying y*y' and noticing that the resulting k^2 term is not equal to the k on the input side. This breaks the linearity condition of adding and multiplying by a constant.
elimenohpee

## Homework Statement

Show that the system is nonlinear:

y*y' + 3y = x

## Homework Equations

if you multiply y * y' , can you merge the y's together to form y^2(t)'? Thats the only way I see this could be nonlinear.

Also the input should be of the same form of the output in a linear system right? For example if my input into my system is of x^2, then the output should also be of y^2. But if the input is x^2 and the output is y^4, then this is considered nonlinear correct?

## The Attempt at a Solution

The book states that for an input x1(t) and x2(t), it should equal the sum of the outputs y1(t) and y2(t). But in the output, the output results in a y^2(t)' , which is not = y(t):

y1y1' + 3y1 = x1
y2y2' + 3y2 = x2

[(y1)^2]' + [(y2)^2]' + 3(y1 + y2) = x1 + x2

Reattempt at solution

I forgot that the system is linear if when you add and also multiply by a constant. So if you multiply by a constant it yields this:

y1k1(y1'k1) + 3y1k1 = x1k1
y2k2(y2'k2) + 3y2k2 = x2k2

but noticing before even adding them together, the k1's and k2's multiply each other in the y*y' function resulting in k^2, which is not equal to the k on the input side (the x side). I'm fairly sure this is why its considered nonlinear, but any input would be great. Thanks again.

(y1)^2 + (y2)^2 + 3(y1 + y2) = x1 + x2
(y1 + y2)^2 + 3(y1 + y2) = x1 + x2
(y^2)' + 3y = x

This system is nonlinear because the output is not directly proportional to the input. The input is of the form x^2, but the output is of the form y^2. In a linear system, the input and output should have the same form in order for the system to be considered linear. In this case, the output is not directly related to the input, making it a nonlinear system. Additionally, the presence of the squared term in the output (y^2) also indicates nonlinearity.

## 1. What is the difference between linear and non-linear systems?

Linear systems have a proportional relationship between the input and output, while non-linear systems do not. In a linear system, the output changes at a constant rate with respect to the input, whereas in a non-linear system, the output changes at a variable rate.

## 2. How do you identify if a system is linear or non-linear?

One way to identify a linear system is by graphing the input and output data and seeing if it forms a straight line. If the graph is not a straight line, then the system is non-linear.

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

A linear system can be represented by a simple pendulum, where the period of the pendulum is directly proportional to the length of the string. A non-linear system can be seen in population growth, where the rate of growth changes as the population increases.

## 4. Can a system be both linear and non-linear?

No, a system can only be either linear or non-linear. It cannot exhibit characteristics of both at the same time.

## 5. What are the applications of linear and non-linear systems in science?

Linear systems are commonly used in physics and engineering to model simple systems such as springs and circuits. Non-linear systems are often used in biology, economics, and social sciences to model complex interactions between variables.

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