$$x(2y+x)y^{\prime \prime }+2x{y^{\prime }}^2+4(x+y)y^{\prime }+2y+x^2=0$$

$$y^{\prime \prime }+{y^{\prime }}^2+1=0$$

$$(x^2+1)y^{\prime \prime }+1+{y^{\prime }}^2=0$$

$$x^3{x}^{\prime \prime }+1=0$$

$$y^{\prime \prime }+{y^{\prime }}^2=1$$

$$y^{\prime \prime }+y{y}^{\prime }=1$$

$$y{y}^{\prime \prime }=1$$

$$x{y}^{\prime \prime }+(6x{y}^2+1)y^{\prime }+2y^3+1=0$$

$$\frac{x{y}^{\prime \prime }}{y+1}+\frac{y{y}^{\prime }-x{y^{\prime }}^2+y^{\prime }}{{(y+1)}^2}=x\sin \left(x\right)$$

$$y{y}^{\prime \prime }\sin \left(x\right)+(y^{\prime }\sin \left(x\right)+\cos \left(x\right)y)y^{\prime }=\cos \left(x\right)$$

$$(\cos \left(y\right)-y\sin \left(y\right))y^{\prime \prime }-{y^{\prime }}^2(2\sin \left(y\right)+y\cos \left(y\right))=\sin \left(x\right)$$

$${y^{\prime \prime }}^2+{y^{\prime }}^2=a^2$$

$$y{y}^{\prime \prime }=1+{y^{\prime }}^2$$

$$y^2{y}^{\prime \prime }=8x^2$$

$$y^{\prime }{y}^{\prime \prime }=1$$

$$x{y}^{\prime \prime }-{y^{\prime }}^2=6x^5$$

$$y^{\prime }{y}^{\prime \prime }=1$$

$$2x{y}^{\prime }{y}^{\prime \prime }={y^{\prime }}^2-1$$

$$x{y^{\prime \prime }}^2+2y=2x$$

$$x^{\prime \prime }+2\sin \left(x\right)=\sin \left(2t\right)$$

$${\mathrm{e}}^{-2t}(y{y}^{\prime \prime }-{y^{\prime }}^2)-2t(t+1)y=0$$

$$t(y{y}^{\prime \prime }+{y^{\prime }}^2)+y{y}^{\prime }=1$$

$$y{y}^{\prime \prime }+{y^{\prime }}^2=1$$

$$y^{\prime \prime }=1+{y^{\prime }}^2$$

$$y{y}^{\prime \prime }=1+{y^{\prime }}^2$$

$$2y{y}^{\prime \prime }=1+{y^{\prime }}^2$$

$$y^3{y}^{\prime \prime }=-1$$

$$y{y}^{\prime \prime }+{y^{\prime }}^2+1=0$$