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