it is ugly, been there done that.
dad, had to put one cow down due to some major problems. young kid i was there.
dad, also had to put a calf that was just born down as well. due to gentical birth defects
been there with a 15 year old dog a few years back as vet put her down.
hit dear on the road. all but one leg completely broken and belly torn up. i waited for a cop to show up. to put the deer out of its misery. if i had a gun i would of done it myself.
have had goldfish live 14 plus years and had to put some of them down over the years. by no means is it easy. but suffering vs being quick. rather it be quick. and if i was in same situation i would want it the same way, quick.
by no means is it easy thing to handle, but it is the right thing.
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for others.
being on a farm. and livestock. i know there are companies that come out to pickup up dead animals.
if ya digging a grave, if ya can get past first few inches, it should be easier to dig after that. the ground shouldn't be as hard from being frozen.
if you live out in the country were there coons, coyotes and like. you will most likely want to dig a few feet deep. and put some rocks or something over it to keep the critters from digging your dog out. been there done that.
also you could get some firewood. and get a good fire going, with some nice coals were grave site will be. and let the red coals burn for a couple hours, then before things cool down to much. pull things over to the side to start digging.