Jakob created the Bernoullian differential equation, y'=p(x)y+q(x)y^n, and the Bernoullian inequality, (1+x)^n>1+nx where x>0.

There was a great amount of rivalry between the brothers. They fought whenever they spoke to each other and began to make ugly critical remarks about each others' work. They also would not appreciate the significance of each others' work.

Johann considered his brother to be a nuisance. He desperately wanted the chair at Basel, but could not have it as long as Jakob was still alive. Instead he accepted an offer to be the chair of mathematics at Groningen. Johan realized shortly before Jakob's death that he was the inferior mathematician, but he would never admit to it. Jakob died August 16, 1705in Basel. Johann became the chair of mathematics at Basel, where he stayed until his death. Some say his quarrelsomeness was matched by his passion for communicating.

At similar times, both the brothers recognized that a sail inflated by the wind is described by (dx/ds)^3=a(d

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