Code p0720 2003 F250 5.4 SD has been stuttering at acceleration between 40 and 55 mph, idle in gear sometimes dies. it showed a p0720 code once and never again although the problem persists. where is the speed control sensor located? Thanks, Jay

By stutter you presumably mean it misfires? Most hesitations and misfires by far are caused by an ignition problem - low output from coils, worn spark plugs and suchlike though bad injectors or low fuel pressure can play a part.

Every diagnostic process should begin with a thorough visual underbonnet inspection and a code read.

sounds like a fuel issue / carb or fuel injection problem.

Has the engine light come on yet?

You'll need to have someone scan the computer for the codes.
Most auto parts stores will scan for free in hopes of selling you some parts. Once you get the codes you can research them online or in this question.

First thing I would do is change the fuel filter and air filter

Have the ODB II codes (computer error codes) pulled and reset. Some auto parts stores will do it for free. Make sure you write them down. You could need new spark plugs and/or wires.

throttle position sensor .. this lets the computer know just where your foot is on the gas .. It gets worn out in that position because that is where it is used the most. I would check it first then go from there... they are priced reasonable .. always start with the least expensive first then go from there. thanks Jerry holler if you need more

Just in Case you Doubt My Solution, It was Straight from the Chilton Repair Manual for your Truck.

Check the catalitic converter in the exaust. It may be stopped up. This can cause these symptoms.

I just went through lots of troubleshooting and expense on my 1996. Whenever I would lift my foot off the accelerator and was below about 30 mph, the rpm's would starts jumping around and sometimes it would stall. If I gave it a little gas (accelerator), I could sometimes keep it running. It also backfired once in a while. Starts right back up. Also, if I let it idle while in part, it would die after 1-2 minutes.

I ended up finding a hole in the fuel regulator. About a ~$70 part if you get an AC Delco, but very simple to change.