How can I repair brown spots in...

Dig out the dead grass and roots with a spade, fill the holes with topsoil, and rake it through. Then spread grass seed over the topsoil as recommended on the package. Rake it lightly and mulch with a thin layer of straw or salt hay. Water well for four to six weeks. Don’t compost the clumps you dug out; they could contain harmful by-products. A better solution is to train your dog to use only one area of your lawn where you have laid out bark mulch, kitty litter, or another absorbent material. Just change the material as needed.