I have a 2003 Lincoln LS v8 The check Transmission light is on, the E synbol instead of D5 or D4 is indicated in the left dash. The car Shifts really hard into second gear after whinding first gear to like 5000 rpms.. Any idea what this may be?

sounds like the fluid level may be low check that first and top up if needed, if its not low does the fluid smell burnt? if it smells the gearbox needs to be checked by transmission specialist. hope that helps you

2002 Lincoln LS Owner Guide 2nd Printing d5 is your drive gear with overdrive the d4 used for loads, towing, pg 164 and 165 explains

use d5 unless your going to be towing and need the assistance from transmission d4 to slow down or tow up hills page 164 and 165 of your owners manual explains your gear selection d5 is overdrive and you get best fuel economy --d4 is for heavy loads towing the following link shows your information
2002 Lincoln LS Owner Guide 2nd Printing

In your Lincoln LS, you will have 3 transmission options to choose from. The first is the SelectShift, which operates similar to a manual transmission but without the clutch. The other 2 are the D4 and D5 options that you are referring to. The number after the D is really a reference to how many gears it will shift through. D4 = 4 gears, D5 = 5 gears. All that being said, it really should operate closer to a 4 gear transmission with an overdrive gear when the car is in D5. Think of it like the "drive" and "overdrive" in other cars. I have heard that to maximize fuel economy, use D5 for highway driving and D4 for city driving.

D5. This will use all 5 gears. D4 will not shift into 5th (overdrive). This causes higher revs on the highway, more noise and less fuel mileage. D4 is only really used when going down long hills for better engine braking. Leave it in d5 and let the cars computer do the shifting for you.

D is for Drive. 5 and 4 is for the gear. the higher the gear the better fuel economy.

E stands for error. Have the vehicle scanned for transmission codes.

transmission shift throttle adjustment under the car in the middle above the drive shaft there is a heat shield remove it and u will see the shift cable

If it is city driving....D4.....Highway....D5

mileage on vehicle?...could be clogged catalytic converter