IOS – How to iPhone restore in iTunes without upgrading iOS

Coincidentally, that happened to me just one week ago when Wifi sync died en route. The solution is simple, but totally non-intuitive: do the exact opposite of what feels right, and all will be well (not exactly Apple’s finest hour in UI intuitiveness):

1. Choose “set up as new phone” on the initial screen. In the dialog that pop ups give your phone the name it had before (important – if your case is similar to mine, it will be pre-filled). This will re-register the phone with iTunes sync.
2. Your phone will do an initial sync with all sync options (music, movies, ebooks etc.) disabled except apps, which are set to “sync all” (if you have a lot of apps in iTunes you are not usually syncing to your phone, the initial sync will flood your phone with these. Delete them out of iTunes beforehand, or from the iPhone after the initial sync). Apps already on the phone will not be overwritten and won’t lose their settings.
3. At this point, all media will still be both in Tunes and on the phone, but not in a synchronized state. Re-check all your sync settings. Acknowledge the “all media on this phone will be deleted” warning. Don’t fret: as long as you are syncing with the same iTunes library as before, nothing will be deleted in the next step.
4. Trigger a sync manually. iTunes will compare what is on the iPhone and what it is meant to sync, and will not re-transfer media already on your phone. Once done, both media libraries will be in sync again. One gotcha though: make sure the downsample setting on the first option page is set as it was before (i.e. on if you downsample music to 128 Kb/s in sync, off otherwise), or you music will be re-synced because bitrates do not match – quite a pain if you have a large library.

After that, your phone will be back ins sync without undue hassle (i.e. worrying about out of date backups in iTunes, iCloud restore etc.).

I'm having the same issue as you described it here. I have a iPhone 5, too.

If you have done anything suggested by Apple, it's a battery issue. You need to change the battery in order it to work. Otherwise, I don't think there is any other way you can make it work. Attached is the link from Apple.

https://support.apple.com/en-us/HT201210

For my situation, before it turned out to error 2001, the temperature of my phone was pretty hot, so I shut it down to let it cool. But after a while, when I wanted to reboot it, it turned out like this.

So I had been searching around the internet for over a hour to seek a solution, all the people were suggesting to try the fix by above link. For me, it did not work. After having reading several forums, they suggested to change the battery. Therefore, I decided to change my battery.
At the same time, I didn't really want to take it to Apple Stores because I think my problem lies on the battery and it would cost me over 70 dollars to let them fix it.

So I ordered the iPhone 5 battery tool kit from Amazon. It was around 15 dollars. I followed a tutorial on the internet to change the battery. It's pretty easy. Then I fixed the problem.