First things first (and I cannot stress this enough) is you need to get your back pain evaluated - there can be many different causes for it and if there is an underlying medical issue you risk further damage. Speak to an appropriate medical professional - doctor or physiotherapist before undertaking any new fitness regime.
If you get the OK from the medical people I would recommend swimming - the body's natural buoyancy means that you are going to be supported, and there's no impact on the joints such as you'll get with running or jumping rope.
Breaststroke is a good starting point for someone with lower back pain - there's minimal trunk rotation and it's a good core workout. Backstroke is another one worth considering if you are confident with it - you have almost no risk of hyper-extending the spine in a backstroke.
It's also key to address the causes of your lower back pain - if you need to study for 13 hours a day you need to address the ergonomics of how you are studying. If you're hunched over a desk/table for thirteen hours straight your back and shoulders are going to hurt!
Ensure that you are sitting in a good posture, take frequent breaks to move around, try to ensure that your exercise includes flexibility work. It doesn't have to take long or require any special equipment - you can even do a seated Yoga Cat-Cow Pose at your desk/workspace!
Source(s): Swimming good for back pain, Breaststroke benefits, and personal experience - my lower back and shoulders have been problem areas for decades but daily swimming (breaststroke) and yoga has made a huge difference.