Solve the Bernoulli equations. xy + y = y -2 A Bernoulli differential equation is of the

Solve the Bernoulli equations.

xy′ + y = y^-2

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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 = y¹-2 = y¹ and du/dx = -y²dy/dx. Then dy/dx = -y² du/dx = -u² du/dx. Substitution into the original equation gives =e^u² -2 du dx - y = exy² du dx +u = −ex. This last equation is linear in the (unknown) dependent variable u.