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

[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.