Using appropriately chosen substitutions, integrate…

Using appropriately chosen substitutions, integrate $y'+y^2-2x^2y+x^4-2x-1=0$.

Step 1: Find the particular solution

Trying $y_1=x^2+1$: $y_1'=2x$, and $2x+(x^2+1)^2-2x^2(x^2+1)+x^4-2x-1$ simplifies to $0$ term by term. ✓

Step 2: Reduce via $y=x^2+1+1/u$

With $Q=2x^2$, $R=-1$: $Q+2Ry_1=2x^2-2(x^2+1)=-2$ (a constant!), giving the simple linear equation $u'-2u=1$.

Step 3: Solve the linear equation

With $\mu(x)=e^{-2x}$: $\dfrac{d}{dx}(e^{-2x}u)=e^{-2x}$, so $e^{-2x}u=-\dfrac12e^{-2x}+C$, i.e. $u=Ce^{2x}-\dfrac12$.