Yes, there is a generalization that covers this case, and much more general second order systems. For example, you can consult L. P. Eisenhart's 1927 book Non-Riemannian Geometry, where he develops the geometry of paths (which is what you are asking about) along the line of his research on the subject with Veblen. He has an effective test (in the form of the vanishing of a certain tensor) for when it is possible (locally) to map the all of the solution curves in $txy$-space to straight lines. I don't have time to compute the tensor in your particular case right now, but it's not hard to do.
By the way, you should be aware that Laguerre's theorem is valid only locally. You may not be able to define such a point transformation globally. For example, the graphs in $ty$-space of the solutions to $y'' + y = 0$ cannot be globally mapped to straight lines because the graphs of solutions can cross each other multiple times.