The following is the general form of a linear ODE, where $t$ is the independent variable and $y$ is the dependent one:
$a_n(t) \frac{d^ny(t)}{dt^n} + a_{n-1}(t) \frac{d^{n-1}y(t)}{dt^{n-1}} + \dots + a_1(t) \frac{dy(t)}{dt} + a_0(t)y(t) = f(t) \cdot g(y)$
and if anyone of the $a_i, i = 0, 1, \dots, n$ coefficients be a function of $y(t)$, it represents a nonlinearity.
But what if a particular case where that is a coefficient that the function form coincides with $y(t)$, (that is $a_i(t) = y(t)$ for some $i$)?
By the way, I've tried to prove that the differential operator $L = a_1(t) \frac{d}{dy} + a_0(y)$ is nonlinear but I've found that it operates on $y(t)$ just as $L = a_1(t) \frac{d}{dy} + a_0(t)$ does. I'm certainly wrong!
May anyone show me the proof? Thanks.
let us take a simple operator $L = \frac{d}{dt} + y$ and look at the equation $$L y = \frac{dy}{dt} - y^2 = 0 \tag 1$$
we can verify that $y_1 = \dfrac{1}{1-t}$ and $y_2 = \dfrac{2}{2-t}$ are solutions of $(1)$ and $y_2(0) = 2y_1(0).$ if $L$ were linear we would have $y_2(t) = 2y_1(t)$ at least on the interval common existence.
do we have that?