Here's why:
* Engine Mounts: These are rubber or hydraulic components that isolate the engine from the chassis. When they wear out or break, the engine becomes less rigidly supported. At idle, the engine's relatively small vibrations are absorbed. However, under load (when driving in gear) or deceleration, the engine's movements are amplified, causing the shaking you experience. The shaking is especially noticeable when slowing down because the engine's momentum is further contributing to the instability. Putting the car in neutral reduces the load on the engine and transmission, lessening the shaking.
* Transmission Mount: A worn or broken transmission mount has a similar effect. It's responsible for supporting the transmission, and a failing mount will allow it to move excessively, leading to vibrations felt throughout the car, particularly when the transmission is under stress (like during acceleration or deceleration).
* Other Possibilities (Less Likely): While less likely given the specific description, other potential causes include:
* Drivetrain Problems: Issues with axles, CV joints, or the differential could cause vibrations, but these would likely be felt more consistently, not just during deceleration.
* Wheel Balance/Alignment: While these can cause vibrations, they usually manifest at higher speeds and are less likely to be dramatically worse only when the engine is under load in gear.
Recommendation:
Have a qualified mechanic inspect your Dodge Neon's engine and transmission mounts. This is the most likely cause based on your description, and it's a relatively straightforward repair. Ignoring this could lead to further damage to other components. Don't continue driving the car excessively with this problem, as it could worsen the condition and potentially lead to more serious damage.