4.38.1 $a{y}^{\prime }(x)+bx{e}^{y(x)}+x{y}^{″}(x)=0$
4.38.2 $(2-a{x}^2)y^{\prime }(x)+x{y}^{″}(x)=0$
4.38.3 $x{y}^{″}(x)=(1-y(x))y^{\prime }(x)$
4.38.4 $x{y}^{″}(x)+x{y}^{\prime }(x{)}^2=y^{\prime }(x)$
4.38.5 $x{y}^{″}(x)=x{y}^{\prime }(x{)}^2+y^{\prime }(x)$
4.38.6 $x{y}^{″}(x)+2x{y}^{\prime }(x{)}^2-2y^{\prime }(x)=0$
4.38.7 $x{y}^{″}(x)=x^2{y}^{\prime }(x{)}^2-2y^{\prime }(x)-y(x{)}^2$
4.38.8 $a{x}^2{y}^{\prime }(x{)}^2+x{y}^{″}(x)+2y^{\prime }(x)=b$
4.38.9 ${(ax{y}^{\prime }(x)-y(x))}^2+x{y}^{″}(x)=b$
4.38.10 $x{y}^{″}(x)=y^{\prime }(x{)}^3+y^{\prime }(x)$
4.38.11 $x{y}^{″}(x)+2y^{\prime }(x)=a{x}^{2k}{y}^{\prime }(x{)}^k$
4.38.12 $2x{y}^{″}(x)+y^{\prime }(x{)}^3+y^{\prime }(x)=0$
4.38.13 $ay(x)(1-y(x{)}^n)+x^2{y}^{″}(x)=0$
4.38.14 $a{e}^{y(x)-1}+x^2{y}^{″}(x)=0$
4.38.15 $(a+1)x{y}^{\prime }(x)+x^2{y}^{″}(x)=x^{k}f(x^{k}y(x),ky(x)+x{y}^{\prime }(x))$
4.38.16 $x^2{y}^{″}(x)+y^{\prime }(x{)}^2=0$
4.38.17 $x^2{y}^{″}(x)=(3x-2y^{\prime }(x))y^{\prime }(x)$
4.38.18 $x^2{y}^{″}(x)+x^2{y}^{\prime }(x{)}^2+4x{y}^{\prime }(x)+2=0$
4.38.19 $x^2{y}^{″}(x)=x^4{y}^{\prime }(x{)}^2-4x^{2}y(x{)}^2+6y(x)$
4.38.20 $a{(x{y}^{\prime }(x)-y(x))}^2+x^2{y}^{″}(x)=b{x}^2$
4.38.21 $a{x}^4{y}^{\prime }(x{)}^2+x^2{y}^{″}(x)+2xy(x)=b$
4.38.22 $ay(x)y^{\prime }(x{)}^2+bx+x^2{y}^{″}(x)=0$
4.38.23 $x^2{y}^{″}(x)=\sqrt{a{x}^2{y}^{\prime }(x{)}^2+by(x{)}^2}$
4.38.24 $x^2{y}^{″}(x)=y(x)f\left(\frac{x{y}^{\prime }(x)}{y(x)}\right)$
4.38.25 $(x^2+1)y^{″}(x)+y^{\prime }(x{)}^2+1=0$
4.38.26 $ay(x{)}^3+9x^2{y}^{″}(x)+2y(x)=0$
4.38.27 $x^3{y}^{″}(x)-x^2{y}^{\prime }(x)=3-x^2$
4.38.28 $x^3(y^{″}(x)+y(x)y^{\prime }(x)-y(x{)}^3)+12xy(x)+24=0$
4.38.29 $x^3{y}^{″}(x)=a{(x{y}^{\prime }(x)-y(x))}^2$
4.38.30 $2x^3{y}^{″}(x)+x^2(2xy(x)+9)y^{\prime }(x)+xy(x)(-2x^{2}y(x{)}^2+3xy(x)+12)-6=0$
4.38.31 $x^4{y}^{″}(x)=x(x^2+2y(x))y^{\prime }(x)-4y(x{)}^2$
4.38.32 $x^4{y}^{″}(x)=x^2{y}^{\prime }(x)(y^{\prime }(x)+x)-4y(x{)}^2$
4.38.33 $x^4{y}^{″}(x)+{(x{y}^{\prime }(x)-y(x))}^3=0$
4.38.34 $x^a{y}^{″}(x)+y(x{)}^b=0$
4.38.35 $(1-12x^2)(3y^{\prime }(x)+y(x{)}^2)+2x(1-4x^2)(y^{″}(x)+y(x)y^{\prime }(x)-y(x{)}^3)-48xy(x)+24=0$
4.38.36 $axy(x)+b-(k{x}^{k-1}-12x^2)(3y^{\prime }(x)+y(x{)}^2)+2(x^k-4x^3)(y^{″}(x)+y(x)y^{\prime }(x)-y(x{)}^3)=0$
4.38.37 $\sqrt{x}{y}^{″}(x)=y(x{)}^{3/2}$
4.38.38 $x^{3/2}{y}^{″}(x)=f\left(\frac{y(x)}{\sqrt{x}}\right)$
4.38.39 $y^{″}(x){(a+2bx+c{x}^2)}^{3/2}=f\left(\frac{x}{\sqrt{a+2bx+c{x}^2}}\right)$
4.38.40 $f(x)f^{\prime }(x)y^{\prime }(x)+f(x{)}^2{y}^{″}(x)=g(y(x),f(x)y^{\prime }(x))$
4.38.41 $f(x{)}^2{y}^{″}(x)=y^{\prime }(x)(3f(x)f^{\prime }(x)-f(x{)}^{2}y(x)+3f(x{)}^3)-24f(x{)}^4$
4.38.42 $f(x{)}^2{y}^{″}(x)=-af(x{)}^5+3f(x)f^{\prime }(x)-f(x{)}^{2}y(x)+3f(x{)}^3$
4.38.43 $2f(x{)}^2{y}^{″}(x)=f(x)y^{\prime }(x)(3f^{\prime }(x)-2f(x)y(x))+f(x)y(x{)}^2{f}^{\prime }(x)+y(x)(f(x)f^{″}(x)-2f^{\prime }(x{)}^2-2f(x{)}^3)+2f(x{)}^{2}y(x{)}^3$
4.38.44 $y(x)y^{″}(x)=a$
4.38.45 $y(x)y^{″}(x)=y^{\prime }(x{)}^2$
4.38.46 $y(x)y^{″}(x)+y^{\prime }(x{)}^2=0$
4.38.47 $y(x)y^{″}(x)=y^{\prime }(x{)}^2-a^2$
4.38.48 $y(x)y^{″}(x)+y^{\prime }(x{)}^2=a^2$
4.38.49 $y(x)y^{″}(x)+y^{\prime }(x{)}^2+y(x{)}^2=0$
4.38.50 $2a^{2}y(x{)}^2+y(x)y^{″}(x)+y^{\prime }(x{)}^2=0$