4.42.1 $y^{\prime }(x)y^{″}(x)=x^{2}y(x)y^{\prime }(x)+xy(x{)}^2$
4.42.2 $(2x^2{y}^{\prime }(x)+y(x{)}^2)y^{″}(x)+2(y(x)+x)y^{\prime }(x{)}^2+x{y}^{\prime }(x)+y(x)=0$
4.42.3 $ay(x{)}^2+x^3{y}^{\prime }(x)y^{″}(x)=0$
4.42.4 $\text{f1}{y}^{\prime }(x)y^{″}(x)+\text{f2}y(x)y^{″}(x)+\text{f3}{y}^{\prime }(x{)}^2+\text{f4}y(x)y^{\prime }(x)+\text{f5}y(x{)}^2=0$
4.42.5 $3y(x)y^{\prime }(x)y^{″}(x)=y^{\prime }(x{)}^3-1$
4.42.6 $(x^2+2y(x{)}^2{y}^{\prime }(x))y^{″}(x)+2y(x)y^{\prime }(x{)}^3+3x{y}^{\prime }(x)+y(x)=0$
4.42.7 $(x-y^{\prime }(x{)}^2)y^{″}(x)=x^2-y^{\prime }(x)$
4.42.8 $(y^{\prime }(x{)}^2+y(x{)}^2)y^{″}(x)+y(x{)}^3=0$
4.42.9 $y^{″}(x)(a(x{y}^{\prime }(x)-y(x))+y^{\prime }(x{)}^2)=b$
4.42.10 $4y(x)y^{\prime }(x{)}^2{y}^{″}(x)=y^{\prime }(x{)}^4+3$
4.42.11 $y^{″}(x)f(y^{\prime }(x))+g(y(x))y^{\prime }(x)+h(x)=0$
4.42.12 $y^{″}(x{)}^2=a+by(x)$
4.42.13 $y^{″}(x{)}^2=a+b{y}^{\prime }(x{)}^2$
4.42.14 $y^{″}(x{)}^2-x{y}^{″}(x)+y^{\prime }(x)=0$
4.42.15 $a^2{y}^{″}(x{)}^2={(y^{\prime }(x{)}^2+1)}^3$
4.42.16 $ax+x{y}^{″}(x{)}^2-2y^{\prime }(x)y^{″}(x)=0$
4.42.17 ${(x{y}^{″}(x)-y^{\prime }(x))}^2=y^{″}(x{)}^2+1$
4.42.18 $2(x^2+1)y^{″}(x{)}^2+2(x-y^{\prime }(x))y^{\prime }(x)-x(4y^{\prime }(x)+x)y^{″}(x)=2y(x)$
4.42.19 $3x^2{y}^{″}(x{)}^2+4y^{\prime }(x{)}^2-2(3x{y}^{\prime }(x)+y(x))y^{″}(x)=0$
4.42.20 $(2-9x)x^2{y}^{″}(x{)}^2+6y(x)y^{″}(x)-6(1-6x)x{y}^{\prime }(x)y^{″}(x)=36x{y}^{\prime }(x{)}^2$
4.42.21 $\text{f0}{y}^{″}(x{)}^2+\text{f1}{y}^{\prime }(x)y^{″}(x)+\text{f2}y(x)y^{″}(x)+\text{g0}{y}^{\prime }(x{)}^2+\text{g1}y(x)y^{\prime }(x)+hy(x{)}^2=0$
4.42.22 $y(x)y^{″}(x)+4y(x)y^{\prime }(x{)}^3-y^{\prime }(x{)}^2=0$
4.42.23 ${(y(x)y^{″}(x)+y^{\prime }(x{)}^2+1)}^2={(y^{\prime }(x{)}^2+1)}^3$
4.42.24 $y^{″}(x{)}^2(a^2-b^{2}y(x{)}^2)+y^{\prime }(x{)}^2(1-b^2{y}^{\prime }(x{)}^2)+2b^{2}y(x)y^{\prime }(x{)}^2{y}^{″}(x)=0$
4.42.25 ${(x^{2}y(x)y^{″}(x)+x^2(-y^{\prime }(x{)}^2)+y(x{)}^2)}^2=4xy(x){(x{y}^{\prime }(x)-y(x))}^3$
4.42.26 $y^{″}(x{)}^3=12y^{\prime }(x)(x{y}^{″}(x)-2y^{\prime }(x))$
4.42.27 $32y^{″}(x){(x{y}^{″}(x)-y^{\prime }(x))}^3+{(2y(x)y^{″}(x)-y^{\prime }(x{)}^2)}^3=0$
4.42.28 $f(y^{″}(x))+x{y}^{″}(x)=y^{\prime }(x)$
4.42.29 $y^{\prime }(x)f\left(\frac{y^{″}(x)}{y^{\prime }(x)}\right)=y^{\prime }(x{)}^2-y(x)y^{″}(x)$
4.42.30 $f(y^{″}(x),y^{\prime }(x)-x{y}^{″}(x),\frac{1}{2}{x}^2{y}^{″}(x)-x{y}^{\prime }(x)+y(x))=0$
4.42.31 $f(x,y^{″}(x))=0$
4.42.32 $f(y(x),y^{″}(x))=0$
4.42.33 $f(y^{\prime }(x),y^{″}(x))=0$
4.42.34 $f(x,y^{\prime }(x),y^{″}(x))=0$
4.42.35 $f(y(x),y^{\prime }(x),y^{″}(x))=0$
4.42.36 $y^{‴}(x)=0$
4.42.37 $y^{‴}(x)=\cos (x)+1$
4.42.38 $y^{‴}(x)+\sin (x)=0$
4.42.39 $y^{‴}(x)={\sin }^3(x)$
4.42.40 $y^{‴}(x)=y(x)$
4.42.41 $y^{‴}(x)=x^2+y(x)$
4.42.42 $y^{‴}(x)=y(x)+e^{x}x+{\cos }^2(x)$
4.42.43 $y^{‴}(x)+ay(x)=0$
4.42.44 $y^{‴}(x)=xy(x)$
4.42.45 $y^{‴}(x)+y^{\prime }(x)=0$
4.42.46 $y^{‴}(x)=y^{\prime }(x)$
4.42.47 $y^{‴}(x)+y^{\prime }(x)=x^3+\cos (x)$
4.42.48 $y^{‴}(x)-2y^{\prime }(x)+4y(x)=0$
4.42.49 $y^{‴}(x)-2y^{\prime }(x)+4y(x)=e^x\cos (x)$
4.42.50 $y^{‴}(x)-3y^{\prime }(x)+2y(x)=0$