The problem boils down to solving a linear system of (first order) differential equations. I suppose you have not had this topic in your math classes yet, so I will not delve into the intricacies. You did the transformation from x and y variables to qs, and now you seem to have what is essentially two uncoupled damped oscillators. I would imagine that you have had the general solution to these explained in your classes. See Wikipedia for the formulae in case you have not: http://en.wikipedia.org/wiki/Linear_differential_equation and  http://en.wikipedia.org/wiki/Damping.

The problem boils down to solving a linear system of (first order) differential equations. I suppose you have not had this topic in your math classes yet, so I will not delve into the intricacies. You did the transformation from x and y variables to qs, and now you seem to have what is essentially two uncoupled damped oscillators. I would imagine that you have had the general solution to these explained in your classes. See Wikipedia for the formulae in case you have not: linear differential equations and damping.