You should treat your lawn against acid from dog urine.
We had same issue with yellow grass after we got our dog. We feed our dog "Lawn Aids" which changes the ph of her urine. It has helped eliminate the yellow grass spots around the lawn. And we've learned that we need to give them to her year round even if there is snow because after stopping one winter we ended up finding lots of yellowed grass in the spring. We've used this product for several years. There are other brands as well. Our dog thinks of it as a doggie snack so there's no issue with getting her to eat them.
As to the yellow grass from the first spring that we had the dog - they fixed themselves.
you can use some old gybroc cut in to small bitts and the cover with fresh soil and lawn seed. the gyproc help neutralize the urine. also works for human urine stains
Awesome solutions in here! I got 2 big fluffy guys at home that I love but their lawn stains are horrible.