Answer: Dry soil makes the grass more susceptible to damage, and certain types of grass are more prone to damage, so that and weather or soil type differences (including fertility and/or moisture holding capacity) may help account for the difference in lawn reactions.
You may want to run some basic soil tests and see if you can add any amendments that would ameliorate the soil. Your county extension should be able to help you with the tests and interpreting the results and you would probably want to do this to get a baseline reading anyway for use in determining your general lawn care program.
It is usually the salts in urine that cause the damage and kill the grass so the best remedy is to water the spot heavily to wash the salts out right away, and rinsing immediately is more effective than waiting until after the urine is dry.
I am not familiar with a soap and baking soda solution for use in this situation, but I suppose you could try a very weak solution (say a teaspoon per gallon) on an experimental basis and see what happens. I suspect any beneficial effect it might have would be due to diluting the urine and rinsing it off.
One other thing to consider is that you might want your vet to test a urine sample and make sure there is nothing unusual going on with your dog's health or diet that would change the content of the urine.
Lastly, sometimes we can train pets to use only a certain area (hopefully an out of the way spot) to do their business and then you would only have a specific limited area to either treat -- or ignore -- as the case may be.
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