Much good advice so far, despite the naysayers.
Only thing I will add is, if you are going to rent a trencher,
PLEASE do yourself a favor and put in a ticket for doing a utilities locate before breaking ground. In FL, you call 811 to initiate a locate ticket [Florida One Call system]; not sure how exactly it works in other states, but you definitely should do this
if you are running any type of powered equipment. "
Call before you dig": it's the motto down here in FL. And pretty much anywhere else in the USA. Also, you have to put the ticket in two full business days before you start the job, to allow time for the utilities to respond and mark their facilities.
OTOH, when it comes to digging in what is presumably the OP's own back yard, the chances he has unknown utilities running
through it are pretty slim. A keen eye can spot where the buried power / telephone / cable / water / sewer lines are routed to supply the neighborhood. Rights-of-way, where utilities may be installed for the neighborhood [as opposed to an individual customer run] are common along the front or back property lines. Less commonly, they may be installed along the sides of a property.
Less obvious, is how the supply is routed to an individual house. These can traverse across the property. Note that your local utility normally won't locate the house supply line as part of their duties. And the utility may not know how this service routes, so you pretty much have to be a well informed property owner. From the FL One Call website:
Note: SSOCOF members typically locate only those underground facilities they own. Water and sewer companies locate main water lines in easements and rights-of-way and lines to a water meter, but may not locate the water facilities from a meter to a home or sewer laterals. Electric companies usually locate their lines to homes and businesses, but not those running to swimming pool pumps and heaters, irrigation systems or another building on the property. Private underground facilities can be located by hiring a contract locator or keeping a detailed drawing of where you placed the private lines.
In situations where utilities are unknown in an excavation area, the safe way to excavate is either by careful hand digging, or by use of a vacuum truck [surely, way beyond the budget of the OP]. As cheap and as lazy as I am, for fifty feet of trench, if I had any idea that there might be something I run into while trenching, I'd be doing it by hand to minimize the impact of possible damage. But it's sandy where I live, and generally easy digging. Maybe less so for the OP...
Lastly, direct bury cable, is indeed intended to be direct buried. The advantage of installing it in PVC is to minimize the possibility of mechanical damage down the line should someone go disturbing the soil with a shovel, trencher, etc.
FWIW, I work for a utility, and regularly locate buried services for our own folks, so that they don't damage our existing facilities ... one of my oldtimer work associates likes to say: "Let's do it right the first time, so we don't have to make excuses later." Wise words indeed.