4.40.1 \(2 y(x) y''(x)=-a^2-4 \left (b-x^2\right ) y(x)^2+y'(x)^2+3 y(x)^4+8 x y(x)^3\)
4.40.2 \(2 y(x) y''(x)=-2 y(x)^2 \left (f'(x)+f(x)^2\right )-3 f(x) y(x) y'(x)+y'(x)^2+8 y(x)^3\)
4.40.3 \(2 y(x) y''(x)=2 x f(x) y(x)^2-4 y(x)^2 y'(x)+y'(x)^2-y(x)^4-1\)
4.40.4 \(2 y(x) y''(x)=3 y'(x)^2\)
4.40.5 \(2 y(x) y''(x)=3 y'(x)^2+4 y(x)^2\)
4.40.6 \(2 y(x) y''(x)=f(x) y(x)^2+3 y'(x)^2\)
4.40.7 \(2 y(x) y''(x)=6 y'(x)^2+\left (1-3 y(x)^2\right ) y(x)^2\)
4.40.8 \(2 y(x) y''(x)=6 y'(x)^2-y(x)^2 \left (a y(x)^3+1\right )\)
4.40.9 \(2 y(x) y''(x)=y'(x)^2 \left (y'(x)^2+1\right )\)
4.40.10 \(3 y(x) y''(x)=2 y'(x)^2+36 y(x)^2\)
4.40.11 \(3 y(x) y''(x)=5 y'(x)^2\)
4.40.12 \(4 y(x) y''(x)=3 y'(x)^2-4 y(x)\)
4.40.13 \(4 y(x) y''(x)=3 y'(x)^2+12 y(x)^2\)
4.40.14 \(4 y(x) y''(x)=a y(x)+b y(x)^2+c y(x)^3+3 y'(x)^2\)
4.40.15 \(5 y(x) y''(x)=y'(x)^2\)
4.40.16 \(12 y(x) y''(x)=15 y'(x)^2-8 y(x)^3\)
4.40.17 \(a y(x) y''(x)=(a-1) y'(x)^2\)
4.40.18 \(a (a+2)^2 y(x) y''(x)=a^2 (a+2) y(x)^3 f'(x)-a^2 f(x)^2 y(x)^4+a (a+2)^2 f(x) y(x)^2 y'(x)+(a-1) (a+2)^2 y'(x)^2\)
4.40.19 \(x y(x) y''(x)+x y'(x)^2+y(x) y'(x)=0\)
4.40.20 \(x y(x) y''(x)+x y'(x)^2=y(x) y'(x)\)
4.40.21 \(x y(x) y''(x)=x y'(x)^2-y(x) y'(x)\)
4.40.22 \(x y(x) y''(x)=x \left (\text {a0}+\text {a1} y(x)^4\right )+y(x) \left (\text {a2}+\text {a3} y(x)^2\right )+x y'(x)^2-y(x) y'(x)\)
4.40.23 \(x y(x) y''(x)+x y'(x)^2+2 y(x) y'(x)=0\)
4.40.24 \(x y(x) y''(x)+x y'(x)^2=3 y(x) y'(x)\)
4.40.25 \(a y(x) y'(x)+f(x)+x y(x) y''(x)+x y'(x)^2=0\)
4.40.26 \(x y(x) y''(x)=a y(x) y'(x)+x y'(x)^2+x y(x)^3\)
4.40.27 \(x y(x) y''(x)=a y(x) y'(x)+b^2 x y(x)^3+x y'(x)^2\)
4.40.28 \(x y(x) y''(x)+2 x y'(x)^2+y(x) y'(x)=0\)
4.40.29 \(x y(x) y''(x)-2 x y'(x)^2+y(x) y'(x)=0\)
4.40.30 \(x y(x) y''(x)-2 x y'(x)^2-y(x) y'(x)=0\)
4.40.31 \(x y(x) y''(x)=2 x y'(x)^2-(y(x)+1) y'(x)\)
4.40.32 \(a y(x) y'(x)+x y(x) y''(x)+2 x y'(x)^2=0\)
4.40.33 \(a y(x) y'(x)+x y(x) y''(x)-2 x y'(x)^2=0\)
4.40.34 \(x y(x) y''(x)-4 x y'(x)^2+4 y(x) y'(x)=0\)
4.40.35 \(a y'(x) \left (x y'(x)-y(x)\right )+x y(x) y''(x)=0\)
4.40.36 \(x (y(x)+x) y''(x)+x y'(x)^2+(x-y(x)) y'(x)=y(x)\)
4.40.37 \(2 x y(x) y''(x)=x y'(x)^2-y(x) y'(x)\)
4.40.38 \(x^2+x (2 y(x)+x) y''(x)+2 x y'(x)^2+4 (y(x)+x) y'(x)+2 y(x)=0\)
4.40.39 \(x^2 y(x) y''(x)+\left (x y'(x)-y(x)\right )^2=0\)
4.40.40 \(x^2 y(x) y''(x)+\left (x y'(x)-y(x)\right )^2=3 y(x)^2\)
4.40.41 \(x^2 y(x) y''(x)=a x y(x) y'(x)+a y(x)^2+2 x^2 y'(x)^2\)
4.40.42 \(a x^2 y'(x)^2+b x y(x) y'(x)+c y(x)^2+x^2 y(x) y''(x)\)
4.40.43 \(x^2 (1-y(x)) y''(x)+2 x^2 y'(x)^2-2 x (1-y(x)) y'(x)+2 (1-y(x))^2 y(x)=0\)
4.40.44 \(x^2 (x-y(x)) y''(x)=\left (x y'(x)-y(x)\right )^2\)
4.40.45 \(x^2 (x-y(x)) y''(x)+\left (x y'(x)-y(x)\right )^2=0\)
4.40.46 \(x^2 (x-y(x)) y''(x)=a \left (x y'(x)-y(x)\right )^2\)
4.40.47 \(2 x^2 y(x) y''(x)=x^2 y'(x)^2-y(x)^2\)
4.40.48 \(2 x^2 y(x) y''(x)=x^2 y'(x)^2+2 x y(x) y'(x)-4 y(x)^2\)
4.40.49 \(x^3 y(x) y''(x)+x^3 y'(x)^2+6 x^2 y(x) y'(x)+3 x y(x)^2=a\)
4.40.50 \(x (x+1)^2 y(x) y''(x)=a (x+2) y(x)^2-2 \left (x^2+1\right ) y(x) y'(x)+x (x+1)^2 y'(x)^2\)