Find the general integral of the Bernoulli equation y…

Find the general integral of the Bernoulli equation $y\,dy-a\dfrac{y^2}{x^2}\,dx=b\dfrac{dx}{x^2}$.

Explanation of Method

Since $y\,dy=\tfrac12 d(y^2)$, substitute $z=y^2$ to reduce directly to a linear equation for $z(x)$.

Step 1: Substitute $z=y^2$

Dividing by $dx$: $yy'=\dfrac{ay^2+b}{x^2}$. With $z=y^2$, $z'=2yy'$: $\dfrac12z'=\dfrac{az+b}{x^2}$, i.e. $z'-\dfrac{2a}{x^2}z=\dfrac{2b}{x^2}$.

Step 2: Solve the linear equation

With $\mu(x)=e^{2a/x}$: $\dfrac{d}{dx}\!\left(e^{2a/x}z\right)=\dfrac{2b}{x^2}e^{2a/x}$. Substituting $u=2a/x$, the right side integrates to $-\dfrac{b}{a}e^{2a/x}+C$, so $z=-\dfrac{b}{a}+Ce^{-2a/x}$.

Result

Final answer:

\[\boxed{\,y^2=Ce^{-2a/x}-\frac{b}{a},\qquad C\in\mathbb{R}}\]