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Sometimes if a dog has had trauma to the rectal area such as continued anal gland issues, fecal incontinence will develop as a result of a loss of muscle tone in the sphincter. This can also be caused by an injury to the back or a problem with an invertebral disc. An intervertebral disc that has slipped or ruptured up into the spinal canal causes inflammation of the spinal cord, which in severe cases causes paralyses of the rear legs. Beagles are prone to disc issues so this may be a possible cause to the fecal leakage. You can read about this here:
You may want to have your dog checked over by your vet as back issues are best treated early for the highest chance of successful treatment. A few things that can help with this are adding fiber to the diet to firm up the feces. A few spoonfuls of canned pumpkin (not pie filling) added to the food works well for this purpose. Also there is one drug that can be prescribed which is phenylpropanolamine which can help tighten the sphincter muscle. It works better on smooth muscles like the bladder but still may help the rectal tissue according to some experts.
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You might want to use a diaper on him until you can get him looked. If it is a disc issue, your vet may give an antiinflammatory medication such as prednisone which will help reduce inflammation and possibly correct the situation. I also would recommend strict crate rest to give any disc injury time to heal with out being aggravated by moving around.
I hope this information is helpful to you. If you would like any additional information or have more questions please don’t hesitate to press the reply to expert or continue conversation button so I can address any issues you still have . If you do find this helpful, please take this opportunity to rate my answer positively so I am compensated for my time.