4.12 Problems 551 to 600

   4.12.1 \(x (2 y(x)+x) y'(x)+(2 x-y(x)) y(x)=0\)
   4.12.2 \(x (x-2 y(x)) y'(x)+(2 x-y(x)) y(x)=0\)
   4.12.3 \(x (-2 y(x)+x+1) y'(x)+y(x) (y(x)-2 x+1)=0\)
   4.12.4 \(x (-2 y(x)-x+1) y'(x)+y(x) (y(x)+2 x+1)=0\)
   4.12.5 \(2 x \left (2 x^2+y(x)\right ) y'(x)+y(x) \left (12 x^2+y(x)\right )=0\)
   4.12.6 \(-3 x^2+2 (x+1) y(x) y'(x)+y(x)^2+2 x=0\)
   4.12.7 \(x (3 y(x)+2 x) y'(x)=y(x)^2\)
   4.12.8 \(x (3 y(x)+2 x) y'(x)+3 (y(x)+x)^2=0\)
   4.12.9 \(\left (x^2+6 x y(x)+3\right ) y'(x)+3 y(x)^2+2 x y(x)+2 x=0\)
   4.12.10 \(x^3+3 x (2 y(x)+x) y'(x)+3 y(x) (y(x)+2 x)=0\)
   4.12.11 \(a x y(x) y'(x)=x^2+y(x)^2\)
   4.12.12 \(a x y(x) y'(x)+x^2-y(x)^2=0\)
   4.12.13 \(x y'(x) (a+b y(x))=c y(x)\)
   4.12.14 \(x (x-a y(x)) y'(x)=y(x) (y(x)-a x)\)
   4.12.15 \(y'(x) (x (A x+B y(x))+\text {a0}+\text {a1} x+\text {a2} y(x))=y(x) (A x+B y(x))+\text {b0}+\text {b1} x+\text {b2} y(x)\)
   4.12.16 \(y'(x) (x (\text {a1}+\text {b2} x+\text {c2} y(x))+\text {a1}+\text {b1} x+\text {c1} y(x))=y(x) (\text {a2}+\text {b2} x+\text {c2} y(x))+\text {a3}+\text {b3} x+\text {c2} y(x)\)
   4.12.17 \(x y'(x) \left (a y(x)+x^n\right )+y(x)^2 (b+c y(x))=0\)
   4.12.18 \(\left (1-x^2 y(x)\right ) y'(x)-x y(x)^2+1=0\)
   4.12.19 \(\left (1-x^2 y(x)\right ) y'(x)+x y(x)^2-1=0\)
   4.12.20 \(x (1-x y(x)) y'(x)+y(x) (x y(x)+1)=0\)
   4.12.21 \(x (x y(x)+2) y'(x)=2 x^3-x y(x)^2-2 y(x)+3\)
   4.12.22 \(x (2-x y(x)) y'(x)-x (x y(x)+1) y(x)^2+2 y(x)=0\)
   4.12.23 \(x (3-x y(x)) y'(x)=y(x) (x y(x)-1)\)
   4.12.24 \(x^2 (1-y(x)) y'(x)+(1-x) y(x)=0\)
   4.12.25 \(x^2 (1-y(x)) y'(x)+(x+1) y(x)^2=0\)
   4.12.26 \(\left (x^2+1\right ) y(x) y'(x)+x \left (1-y(x)^2\right )=0\)
   4.12.27 \(\left (1-x^2\right ) y(x) y'(x)+2 x^2+x y(x)^2=0\)
   4.12.28 \(2 x^2 y(x) y'(x)=x^2 (2 x+1)-y(x)^2\)
   4.12.29 \(x (1-2 x y(x)) y'(x)+y(x) (2 x y(x)+1)=0\)
   4.12.30 \(x (2 x y(x)+1) y'(x)+y(x) (3 x y(x)+2)=0\)
   4.12.31 \(y(x) \left (-x^2 y(x)^2+2 x y(x)+1\right )+x (2 x y(x)+1) y'(x)=0\)
   4.12.32 \(x^2 (x-2 y(x)) y'(x)=2 x^3-4 x y(x)^2+y(x)^3\)
   4.12.33 \(2 x (x+1) y(x) y'(x)=y(x)^2+1\)
   4.12.34 \(3 x^2 y(x) y'(x)+2 x y(x)^2+1=0\)
   4.12.35 \(x^2 (4 x-3 y(x)) y'(x)=y(x) \left (6 x^2-3 x y(x)+2 y(x)^2\right )\)
   4.12.36 \(\left (1-x^3 y(x)\right ) y'(x)=x^2 y(x)^2\)
   4.12.37 \(a+2 x^3 y(x) y'(x)+3 x^2 y(x)^2=0\)
   4.12.38 \(x \left (3-2 x^2 y(x)\right ) y'(x)=3 x^2 y(x)^2-3 y(x)+4 x\)
   4.12.39 \(x \left (2 x^2 y(x)+3\right ) y'(x)+y(x) \left (3 x^2 y(x)+4\right )=0\)
   4.12.40 \(3 x^4+8 x^3 y(x) y'(x)-6 x^2 y(x)^2-y(x)^4=0\)
   4.12.41 \(x y(x) \left (a+b x^2\right ) y'(x)=A+B y(x)^2\)
   4.12.42 \(3 x^4 y(x) y'(x)=1-2 x^3 y(x)^2\)
   4.12.43 \(x^7 y(x) y'(x)=5 x^3 y(x)+2 \left (x^2+1\right )\)
   4.12.44 \(\sqrt {x^2+1} y(x) y'(x)+x \sqrt {y(x)^2+1}=0\)
   4.12.45 \(\sqrt {x^2+1} (y(x)+1) y'(x)=y(x)^3\)
   4.12.46 \(y'(x) (\text {g0}(x)+\text {g1}(x) y(x))=\text {f0}(x)+\text {f1}(x) y(x)+\text {f2}(x) y(x)^2+\text {f3}(x) y(x)^3\)
   4.12.47 \(y(x)^2 y'(x)+x (2-y(x))=0\)
   4.12.48 \(y(x)^2 y'(x)=x \left (y(x)^2+1\right )\)
   4.12.49 \(\left (y(x)^2+x\right ) y'(x)+y(x)=a+b x\)
   4.12.50 \(\left (x-y(x)^2\right ) y'(x)=x^2-y(x)\)