Solve the Bernoulli equations.

x²y′ + 2xy = y³

HISTORICAL BIOGRAPHY James Bernoulli (1654-1705) A Bernoulli differential equation is of the form dy dx Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u = y-n transforms the Bernoulli equation into the linear equation du dx For example, in the equation + P(x)y= Q(x)y". or, equivalently, + (1 n)P(x)u = (1 n)Q(x). dy dx we have n = 2, so that u = y-2 = y and du/dx = -y- dy/dx. Then dy/dx = -y du/dx = u du/dx. Substitution into the original equation gives u = ex u -u -2 du dx du dx y = exy +u = ex. This last equation is linear in the (unknown) dependent variable u.