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