The Newton's notation could be a bit confusing sometimes.
Besides the easiness to solve, are there other reasons why we might prefer linear over non-linear or vice versa?
yeah its better to write whose derivative it is. By the way, Burger's equation is a shockwave simulation function. the one here is the general form called viscous burger equation. For ex, a thin ideal pipe with fluid running through it, Burgers' equation describes the speed of the fluid at each location along the pipe as time progresses.
Is the u here position, velocity, or something else?