Graphical examination for existence of 2nd order ODE given a solution curve Given any solution curve $y(t)$, what are some graphical criterion to determine whether there exists some 2nd order linear, homogeneous, with continuous but possibly non-constant coefficients, to which the curve is a solution? 
Each of the plots below show graphs of three functions $y(t)$. For which options do there exist a second order linear homogeneous ode with continuous but possibly non-constant coefficients to which all three functions are solutions of?


The answers are (a), (c), (f). 
Notice that the red graph $y_3$ of option (f) is exactly in the middle, so it is a linear combination of $y_1$ and $y_2$.In comparison of option (e), the red graph is not a linear combination of the other two, so we get three linearly independent solutions. But the solution space of a second order ODE has dimension two, so option (e) is not possible.
In option (d), the green curve and the red curve have common zeros, which is not allowed unless they are constant multiples of each other. If they were constants multiples of each other, they would have the same zeros.
*Please view my other answer for why option (b) is incorrect. 
Therefore options (b), (d) and (e) are incorrect.
