Definition linear ODE An ordinary differential equation is said to be linear if 
$$F(t,y(t),...,y^{(n)}(t))=0$$
is linear in every derivative. I run into a little problem when using this definition for the equation
$$y'y=0$$
because we have that $$F(t,\alpha y_1(t)+\beta y_2(t),y'(t))=(\alpha y_1(t)+\beta y_2(t))y'(t)=\alpha y_1(t)y't(t) + \beta y_2(t)y'(t)\\
=\alpha F(t,y_1(t),y'(t)) + \beta F(t,y_2(t),y'(t))$$
and vice versa for $y'(t)$. This shows that the ODE is linear. But an equivalent definition of linearity states that it has to have the form
$F(t,y(t),...,y^{(n)}(t))=\sum_{k=0}^n a_i(t)y^{(i)}(t) - g(t)=0$
With this definition the ODE is not linear anymore. Not sure what my mistake is there.

We have $F(t,y(t),y’(t))=yy’$. Note that $$F(\alpha t, \alpha y(t), \alpha y’(t)) =\alpha^2yy’ \neq \alpha yy’=\alpha F(t,y(t),y’(t))$$
and thus $F$ is not linear in its arguments, implying that it is not a linear differential equation using your first definition. 
