Shoring up drainage culvert slope
Your pictures appear to show a steeply sloped section with erosion, a flat section, and another sloped section (without erosion.) If the flat section resulted from poor compaction of an original sloped profile that did not have the flat part, you may be able to simply fill that back in so the slope is not as steep. If you want to keep it in mown grass, that might be the best bet.

Otherwise, "rip-rap" or fairly large rocks are arguably the standard method of holding a steep drainage slope.