Bernoulli Equations

A Bernoulli equation has the form \frac{dy}{dx} + P(x)y = Q(x)y^n with n \neq 0,1, and the key strategy is the substitution v = y^{1-n}, which linearizes it. The underlying reason is that dividing through by y^n and introducing v turns the nonlinear y^n term into a linear term in v. Here n = 2, so v = y^{1-2} = y^{-1}, and differentiating gives \frac{dv}{dx} = -y^{-2}\frac{dy}{dx}. Dividing the original equation by y^2 yields y^{-2}\frac{dy}{dx} + y^{-1} = x. Substituting -\frac{dv}{dx} for y^{-2}y' and v for y^{-1} gives -\frac{dv}{dx} + v = x, which rearranges to \frac{dv}{dx} - v = -x, a standard linear equation. Option v = y^2 uses the wrong exponent for n = 2. Option v = y^{-1} with \frac{dv}{dx} + v = x has the sign of the v term wrong because it ignores the negative from differentiating y^{-1}. Option v = \ln y applies only to a different equation type. This matches the JEE Advanced Bernoulli reduction. As a final check, the resulting equation is genuinely linear in v with integrating factor e^{-x}, confirming the reduction succeeded.