4.19.1 \(a^2+2 a x y'(x)-2 a y(x)+x^2 y'(x)^2+x^2=0\)
4.19.2 \(x^2 y'(x)^2-2 x y(x) y'(x)+y(x) (y(x)+1)-x=0\)
4.19.3 \(-x^4+x^2 y'(x)^2+\left (1-x^2\right ) y(x)^2-2 x y(x) y'(x)=0\)
4.19.4 \(x^2 y'(x)^2-(2 x y(x)+1) y'(x)+y(x)^2+1=0\)
4.19.5 \(-(a+2 x y(x)) y'(x)+x^2 y'(x)^2+y(x)^2=0\)
4.19.6 \(x^2 y'(x)^2-x (x-2 y(x)) y'(x)+y(x)^2=0\)
4.19.7 \(-4 a+x^2 y'(x)^2+2 x (y(x)+2 x) y'(x)+y(x)^2=0\)
4.19.8 \(x \left (x^3-2 y(x)\right ) y'(x)-\left (2 x^3-y(x)\right ) y(x)+x^2 y'(x)^2=0\)
4.19.9 \(x^2 y'(x)^2+3 x y(x) y'(x)+2 y(x)^2=0\)
4.19.10 \(x^3+x^2 y'(x)^2-3 x y(x) y'(x)+2 y(x)^2=0\)
4.19.11 \(x^2 y'(x)^2+4 x y(x) y'(x)-5 y(x)^2=0\)
4.19.12 \(x^2 y'(x)^2-4 x (y(x)+2) y'(x)+4 y(x) (y(x)+2)=0\)
4.19.13 \(x^2 y'(x)^2-5 x y(x) y'(x)+6 y(x)^2=0\)
4.19.14 \(x^2 y'(x)^2+x \left (x^2+x y(x)-2 y(x)\right ) y'(x)+(1-x) \left (x^2-y(x)\right ) y(x)=0\)
4.19.15 \(x^2 y'(x)^2+(y(x)+2 x) y(x) y'(x)+y(x)^2=0\)
4.19.16 \(x^2 y'(x)^2+(2 x-y(x)) y(x) y'(x)+y(x)^2=0\)
4.19.17 \(y'(x) \left (a+b x^2 y(x)^3\right )+a b y(x)^3+x^2 y'(x)^2=0\)
4.19.18 \(\left (1-x^2\right ) y'(x)^2=1-y(x)^2\)
4.19.19 \(\left (1-x^2\right ) y'(x)^2+4 x^2+2 x y(x) y'(x)=0\)
4.19.20 \(\left (a^2+x^2\right ) y'(x)^2=b^2\)
4.19.21 \(\left (a^2-x^2\right ) y'(x)^2+b^2=0\)
4.19.22 \(\left (a^2-x^2\right ) y'(x)^2=b^2\)
4.19.23 \(\left (a^2-x^2\right ) y'(x)^2=x^2\)
4.19.24 \(\left (a^2-x^2\right ) y'(x)^2+x^2+2 x y(x) y'(x)=0\)
4.19.25 \(\left (a^2-x^2\right ) y'(x)^2-2 x y(x) y'(x)-y(x)^2=0\)
4.19.26 \(\left (a^2+x^2\right ) y'(x)^2+b-2 x y(x) y'(x)+y(x)^2=0\)
4.19.27 \(\left (2 x^2+1\right ) y'(x)^2+\left (x^2+2 x y(x)+y(x)^2+2\right ) y'(x)+2 y(x)^2+1=0\)
4.19.28 \(4 x^2 y'(x)^2-4 x y(x) y'(x)=8 x^3-y(x)^2\)
4.19.29 \(a x^2 y'(x)^2+(1-a) a x^2-2 a x y(x) y'(x)+y(x)^2=0\)
4.19.30 \(\left (1-a^2\right ) x^2 y'(x)^2-a^2 x^2-2 x y(x) y'(x)+y(x)^2=0\)
4.19.31 \(x^3 y'(x)^2=a\)
4.19.32 \(x^3 y'(x)^2+x y'(x)-y(x)=0\)
4.19.33 \(a+x^3 y'(x)^2+x^2 y(x) y'(x)=0\)
4.19.34 \(x^3 y'(x)^2-\left (2 x^2 y(x)+1\right ) y'(x)+x y(x)^2=0\)
4.19.35 \(x \left (1-x^2\right ) y'(x)^2-2 \left (1-x^2\right ) y(x) y'(x)+x \left (1-y(x)^2\right )=0\)
4.19.36 \(4 x (a-x) (b-x) y'(x)^2=\left (-2 x (a+b)+a b+2 x^2\right )^2\)
4.19.37 \(x^4 y'(x)^2-x y'(x)-y(x)=0\)
4.19.38 \(x^4 y'(x)^2+2 x^3 y(x) y'(x)-4=0\)
4.19.39 \(x^4 y'(x)^2+x y(x)^2 y'(x)-y(x)^3=0\)
4.19.40 \(x^2 \left (a^2-x^2\right ) y'(x)^2+1=0\)
4.19.41 \(3 x^4 y'(x)^2-x y(x)-y(x)=0\)
4.19.42 \(4 x^5 y'(x)^2+12 x^4 y(x) y'(x)+9=0\)
4.19.43 \(x^6 y'(x)^2-2 x y'(x)-4 y(x)=0\)
4.19.44 \(x^8 y'(x)^2+3 x y'(x)+9 y(x)=0\)
4.19.45 \(y(x) y'(x)^2=a\)
4.19.46 \(y(x) y'(x)^2=a^2 x\)
4.19.47 \(y(x) y'(x)^2=e^{2 x}\)
4.19.48 \(2 a x y'(x)-a y(x)+y(x) y'(x)^2=0\)
4.19.49 \(-4 a^2 x y'(x)+a^2 y(x)+y(x) y'(x)^2=0\)
4.19.50 \(a x y'(x)+b y(x)+y(x) y'(x)^2=0\)