How do I get rid of spotted knapweed without spraying?

You can dig them:

1. Pull plants during their bloom period and before they’ve gone to seed. Be sure to bag and dispose of the plants in a landfill, to avoid dispersing seed or root fragments. A good rain just before (or even during) pulling makes the job much easier.
2. Be patient when pulling and learn to stop and dig more when you feel that the root is about to snap. Pieces of root left in the ground will sprout the following year.

Don’t dig right at the base of the plant – you risk cutting through the root, which often twists to one side or another a few inches down. Instead, loosen the soil about 5-6 inches from the base of the plant.

I dig the spotted knapweed in areas where there isn't much, but I have 10 acres and some of it is solid. My neighbor doesn't take care of his spotted knapweed, so I have plenty of new plants. I'd like a way to spray with a natural spray instead of Weed killer like Stop Weed (which is actually pretty good because it doesn't kill my good plants).

Rent some goats to eat the weeds! Then use mulch to smother the roots. Warning: this will kill ALL the plants in the area. You will need to replant after taking up the mulch covering.