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?


Not a complete answer, but too long for a comment.
Upon closer inspection, I'll concede that (b) is in fact not a solution of the original question.
Note that the solution curves in (b) all differ by a constant. If there exists a second-order ODE that admits all of these curves as a solution, then it must also admit a constant as a solution.
In general, an IVP that admits a constant solution looks something like this

$$ y''(t) + p(t)y'(t) = 0, \quad y(t_0) = y_0, \ y'(t_0) = 0 \tag{1} $$

where $p(t)\ne 0$. 
If the ODE in $(1)$ has a non-constant solution, said solution must not have a zero derivative at any point (since if a solution has a zero derivative, it must be constant due to the Uniqueness Theorem). We see that the non-constant curves in (b) do not satisfy this, therefore (b) is not correct.
