Find the general integral of x^2y'=x^2y^2+xy+1 , given that…

Find the general integral of $x^2y'=x^2y^2+xy+1$, given that a particular integral has the form $y=\dfrac{a}{x}$.

Step 1: Find $a$

Substituting $y=a/x$: $-a=a^2+a+1\Rightarrow(a+1)^2=0\Rightarrow a=-1$, so $y_1=-1/x$.

Step 2: Reduce and solve for $u$

With $y=-\dfrac1x+\dfrac1u$, substitution into $y'=y^2+\dfrac{y}{x}+\dfrac{1}{x^2}$ gives $u'-\dfrac{u}{x}=-1$. With $\mu(x)=1/x$: $\dfrac{d}{dx}\!\left(\dfrac{u}{x}\right)=-\dfrac1x$, so $\dfrac{u}{x}=-\ln|x|+C$, i.e. $u=x(C-\ln|x|)$.