Inspiron 5502/5509 Power adapter not recognized

Hi!

Sounds like you have got a failing AC adapter. I would suggest you to run a full diagnostic test just to err on the safe side, this way we'll make sure there are no other issues involved: on startup, as soon as you see the Dell splash screen start tapping F12 to enter the pre-boot menu, where you will find different booting options and the Diagnostics tool: run a full diagnostic test.

Then let's make sure the battery is still in a good shape:

1. in Start type cmd to find the Command Prompt, and run it as administrator;
2. paste in the following command and hit enter:

   powercfg /batteryreport

   
   
3.  in C:\Windows\System32 you will find a .html report that you can open with any web browser: grab a screenshot and post it here

IF everything else can be excluded, then it's unfortunately time to look into replacing the AC adapter. If your laptop is still under warranty (did you buy a warranty extension?) you can have it replaced for free.

Otherwise you will have to buy one yourself. It's a Dell 4.5mm barrel charger, 45W or 65W depending on the configuration of your pc, but you can choose any wattage you like as long as it exceeds the older adapter's power: if your current charger is rated 45W, you can buy a 65W, even 180W (though it's likely heavier to carry); if your current charger is rated at 65W, you cannot buy a 45W charger, you need it at least to be 65W and up.

Just be careful when you shop for a new adapter, choose only reputable sellers, there are a lot of fake or refurbished adapters on the market, eBay, Amazon and Aliexpress are littered with them.
You have posted on the Italian community forum, so I would guess you are currently living in Italy? dell.com offers a 65W 4.5 barrel at a decent price (shipping is included). Or a 45W.

A reputable online reseller, IPC Computer, sells the 65W at EUR 48 + shipping just to give you a comparison.