Solve the Bernoulli equations

y′ - y = -y²

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