Linear and non linear systems

[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

(added together)

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

 

[elimenohpee]

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.

 

[Mene]

(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.