Consider the following nonlinear ODE:
$$m\ddot s=s-s^2$$
Suppose $s_1(t)$ and $s_2(t)$ are solutions and $k_1$ and $k_2$ are constants. Is $k_1s_1(t)$ a solution?
How would one check this?
Also, the answer says that in general superposition does not hold for nonlinear ODE's. What is meant by this?
$k_1s_1(t)$ is not solution (of course with $k_1\neq 1$ and $k_1\neq 0$).
To check it, put it into the ODE and obseve if it agrees or not. :
$m(k_1s_1)''\neq (k_1s_1)-(k_1s_1)^2$ because $ms_1''\neq s_1-k_1s_1^2$ , since $s_1''= s_1-s_1^2$. So, it doesn't agree.
Superposition does not hold for nonlinear ODE's means that the linear combination of particular solutions is not solution of the ODE.
For example $k_1s_1(t)+k_2s_2(t)$ is not solution of the ODE. Put it into the ODE and observe that it doesn't agree.