* Differential Fluid Leak: A leak in the differential housing on the driver's side is the most likely culprit. The shaking is caused by insufficient lubrication, leading to friction and potential damage within the differential. The burning smell is the fluid overheating and possibly burning. Look carefully under the car on the driver's side for evidence of fluid dripping or pooling. Check the fluid level in the differential (if accessible).
* Differential Bearing Failure: A failing bearing within the differential, particularly on the driver's side, can also cause shaking and heat generation, leading to a burning smell. This is more serious than a simple leak and will require professional attention.
* Axle Seal Failure (Driver's Side): A worn or damaged axle seal on the driver's side could leak differential fluid, causing the same symptoms.
* Brake Caliper or Pad Issue (Less Likely): While less likely given the focus on the smell and shaking *in reverse*, a seized or sticking brake caliper on the driver's side could cause the shaking. However, a burning smell from brakes typically has a more distinct, acrid odor.
What to do:
1. DO NOT drive the car further: Continuing to drive with a suspected differential problem can cause significant and costly damage.
2. Visually inspect: Carefully examine the driver's side of the rear differential and axle for leaks or damage. Look for fluid trails, pools of fluid, or any signs of unusual wear.
3. Check the differential fluid level (if accessible): Your Grand Prix's owner's manual will show you where the differential fill plug is located. If the fluid is low or smells burnt, this confirms a problem.
4. Professional inspection: Take your Grand Prix to a qualified mechanic for diagnosis and repair. Describing the symptoms (shaking in reverse, burning smell, driver's side only) will help them pinpoint the issue. They will have the tools and expertise to accurately assess the damage and determine the best course of action, which might involve a differential fluid change, seal replacement, bearing replacement, or more extensive repairs.
Ignoring this problem could lead to complete differential failure, requiring a costly replacement. Get it checked out as soon as possible.
