LED tv LG 50LN5200 with no picture but has sound

Chances are it's bad backlights if you can see the image with a flashlight

The backlight leds need replacing, you can get replacements very easy around $40 a set however it is a tricky job if you have not done it before, the lcd panel must be removed without damage, the backlights are stuck to the metal cover behind lcd panel.

Likely the diodes are also blown, and are what caused the capacitors to blow.

If you saw no faint picture with the flashlight, then it's the T-con board.

If you can't see a faint picture with the flashlight then the T-con board is gone. If you can see a faint picture, then the led driver or ballasts for the back lights quit. You can get parts from here: TV Parts Universal TV Stands DLP Chips ShopJimmy

Check the following link and watch the video from the other link provided in that answer. It has a guide to isolate the failure to a particular board.
have LG 47in tv sound but no picture and can see an image Replaced the...

If you wish to get some details; check the site linked here. Pull up older posts.Surf the site with patience.http://electronicshelponline.blogspot.com/
You can repair it very easily, within seconds.

power board, if you can find one for it

usually bad power supply, Where did you source your part? return/exchange it. Bad LEDs/ possible but not as likely, Tcon working if you can see images with flashlight. You must also have backlight on signal from main board ,check for 3 to 5v on labeled connecter on power supply. UPDATE---47" & 50" LN series. This model has had LED problems if your TV acts as described by Mr. Hughes in is answer to this problem you have bad LEDs

Try the old flashlight test, turn off the lights in the room. Turn on the set and flash a bright flashlight on the screen. Do you see the image or is the screen totally black?? If you see the image, then most likely you have a backlight problem. Verify that you have 24V on the power supply for the inverter