4.39.1 \(y(x) y''(x)=\text {a0}+\text {a1} y(x)+y(x)^3 (\text {a2}+\text {a3} y(x))+y'(x)^2\)
4.39.2 \(y(x) y''(x)=\text {a0}+\text {a1} y(x)+\text {a2} y(x)^2+\text {a3} y(x)^3+\text {a4} y(x)^4+y'(x)^2\)
4.39.3 \(y(x) y''(x)=y'(x)^2+y(x) y'(x)\)
4.39.4 \(y(x) y''(x)=e^x y(x) \left (\text {a0}+\text {a1} y(x)^2\right )+e^{2 x} \left (\text {a2}+\text {a3} y(x)^4\right )+y'(x)^2\)
4.39.5 \(y(x) y''(x)=y'(x)^2+y(x)^2 \log (y(x))\)
4.39.6 \(y(x) y''(x)=-x^2 y(x)^2+y'(x)^2+y(x)^2 \log (y(x))\)
4.39.7 \(y(x) y''(x)+y'(x)^2=y'(x)\)
4.39.8 \(y(x) y''(x)=y'(x)^2-y'(x)\)
4.39.9 \(y(x) y''(x)=y(x)^2 \left (f(x) y(x)+g'(x)\right )+y'(x)^2+y'(x)\)
4.39.10 \(y(x) y''(x)=y'(x)^2-2 y'(x)\)
4.39.11 \(y(x) y''(x)+y'(x)^2-x y'(x)+y(x)=0\)
4.39.12 \(a x y'(x)+y(x) y''(x)+y'(x)^2=0\)
4.39.13 \(y(x) y''(x)=-y(x) f'(x)+f(x) y'(x)+y'(x)^2+y(x)^3\)
4.39.14 \(y(x) y''(x)=y(x) f''(x)-f(x) y'(x)-f(x) y(x)^3+y'(x)^2+y(x)^4\)
4.39.15 \(y(x) y''(x)=-a y(x) y'(x)-b y(x)^2+y'(x)^2\)
4.39.16 \(y(x) y''(x)=a y(x) y'(x)+b y(x)^2+y'(x)^2+y(x)^3\)
4.39.17 \(y(x) y''(x)=y(x)^2 y'(x)+y'(x)^2\)
4.39.18 \(y(x) y''(x)=f(x) y(x) y'(x)+g(x) y(x)^2+y'(x)^2\)
4.39.19 \(y(x) y''(x)=-y(x) \left (f'(x)-y(x)^2 g'(x)\right )+y'(x) \left (f(x)+g(x) y(x)^2\right )+y'(x)^2\)
4.39.20 \(y(x) y''(x)=2 y'(x)^2+y(x)^2\)
4.39.21 \(y(x) y''(x)=2 \left (y'(x)^2-y(x)^2\right )\)
4.39.22 \(y(x) y''(x)=-3 y(x) y'(x)+3 y'(x)^2+y(x)^2\)
4.39.23 \(y(x) y''(x)=a y'(x)^2\)
4.39.24 \(y(x) y''(x)=a y'(x)^2+b\)
4.39.25 \(y(x) y''(x)=a y'(x)^2+b y(x)^3\)
4.39.26 \(y(x) y''(x)=a y'(x)^2+\text {a0}+\text {a1} y(x)+\text {a2} y(x)^2+\text {a3} y(x)^3+\text {a3} y(x)^2+\text {a4} y(x)^4\)
4.39.27 \(y(x) y''(x)=a y'(x)^2+b y(x) y'(x)+c y(x)^2\)
4.39.28 \(y(x) y''(x)=\text {a3} y(x)^{a+1}+a y'(x)^2+\text {a1} y(x) y'(x)+\text {a2} y(x)^2\)
4.39.29 \(a y'(x)^2+f(x) y(x) y'(x)+g(x) y(x)^2+y(x) y''(x)=0\)
4.39.30 \(y(x) y''(x)+y'(x)^3=0\)
4.39.31 \(y(x) y''(x)+y'(x)^3-y'(x)^2=0\)
4.39.32 \(y(x) y''(x)=y'(x)^2 \left (y'(x) (-\sin (y(x)))-y(x) y'(x) \cos (y(x))+1\right )\)
4.39.33 \((1-y(x)) y''(x)+2 y'(x)^2=0\)
4.39.34 \((a+y(x)) y''(x)=y'(x)^2\)
4.39.35 \((a+y(x)) y''(x)+y'(x)^2=b\)
4.39.36 \((a+y(x)) y''(x)+b y'(x)^2=0\)
4.39.37 \((y(x)+x) y''(x)+y'(x)^2-y'(x)=0\)
4.39.38 \((x-y(x)) y''(x)+2 y'(x) \left (y'(x)+1\right )=0\)
4.39.39 \((x-y(x)) y''(x)=\left (y'(x)+1\right ) \left (y'(x)^2+1\right )\)
4.39.40 \((x-y(x)) y''(x)=f\left (y'(x)\right )\)
4.39.41 \(2 y(x) y''(x)=y'(x)^2\)
4.39.42 \(2 y(x) y''(x)+y'(x)^2+1=0\)
4.39.43 \(2 y(x) y''(x)=a+y'(x)^2\)
4.39.44 \(2 y(x) y''(x)=y'(x)^2+8 y(x)^3\)
4.39.45 \(2 y(x) y''(x)=y'(x)^2+8 y(x)^3+4 y(x)^2\)
4.39.46 \(2 y(x) y''(x)=y'(x)^2+4 (2 y(x)+x) y(x)^2\)
4.39.47 \(2 y(x) y''(x)=y(x)^2 (a+b y(x))+y'(x)^2\)
4.39.48 \(2 y(x) y''(x)=a y(x)^3+y'(x)^2-2 x y(x)^2-1\)
4.39.49 \(2 y(x) y''(x)=y(x)^2 (a x+b y(x))+y'(x)^2\)
4.39.50 \(2 y(x) y''(x)=y'(x)^2+3 y(x)^4\)