Consider the following nonlinear second order ordinary differential equation:
$$\ddot y + p(t,y)\dot y + q(t,y)y = 0 $$
Is the above equation autonomous? I would say no, due to the dependence on t
Is the above equation homogeneous? I would say yes, due to the RHS
Suppose two functions $y = \phi _1(t) $ and $ y = \phi _2(t) $ solve the ODE individually. Is $y = a_1\phi _1(t) + a_2\phi _2(t) $ where $a_1$ and $a_2$ are scalar constants, also a solution? I would say no based on the principle of superposition
Now here is my question:
Write the minimum relaxation (smallest possible modification) to the ODE such that $y = a_1\phi _t(t)$ is also a solution of the relaxed form.
Isn't the above a trick question? The solution is simply multiplying a given solution by a constant and the original ODE is homogeneous
Any clarification would be greatly appreciated!