An equation of the form y' + a(x) y = b(x)y" is called a Bernoulli equation, named for Jakob Bernoulli (1654-1705), one of a family of noted Swiss scientists and mathematicians. Note that if n = 0 or n = 1, we have a linear equation. Now if n is not equal to 0 or 1 and we divide both sides of the equation by y", we can let ,1-n and get a linear equation in the variable z. We solve the linear equation for z in terms of x and then return to the original variables x and y. This substitution method was found by Leibniz in 1696. For example, y'- y = xy is a Bernoulli equation with a(x) = 1, b(x) = x, and n = 2. Divide by y2 and the equation be- comes y- y'- y = x. Letting z = y, we get the linear equation -z' -z = x, or z' + z = -x. Solving for z, we find that z = 1 - x + ce*. Since z = =y-, we conclude that y = (1-x+ce)-. Note that we divided by y and that y = 0 is a singular solution. Find solutions for each Bernoulli equation in Problem x = = x + x