There is nothing tough about it, assuming you don't mind eating in diners and staying in motels.

There are car transport companies of two kinds:

1. Drive-away services, where someone will drive the car for you. Cheaper, but you're getting miles on the car, and a stranger driving it.

2. Truck transportation services. More expensive, but more peace-of-mind. In my experience, only available between major cities (but LA-DC shouldn't be difficult).