$$y^{\prime \prime }=x{\mathrm{e}}^x$$

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

$$y^{\prime \prime }y-{\left(y^{\prime }\right)}^2-y^2{y}^{\prime }=0$$

$$y^{\prime \prime }y-{\left(y^{\prime }\right)}^2+1=0$$

$$2y^{\prime \prime }={\mathrm{e}}^y$$

$$y^{\prime \prime }y+2y^{\prime }-{\left(y^{\prime }\right)}^2=0$$

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

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

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

$$x^2{y}^{\prime \prime }+3x{y}^{\prime }+y=x$$

$${(-1+x)}^2{y}^{\prime \prime }+4(-1+x)y^{\prime }+2y=\cos (x)$$

$$(x^3-x)y^{\prime \prime \prime }+(8x^2-3)y^{\prime \prime }+14x{y}^{\prime }+4y=0$$

$$2x^{3}y{y}^{\prime \prime \prime }+6x^3{y}^{\prime }{y}^{\prime \prime }+18x^{2}y{y}^{\prime \prime }+18x^2{\left(y^{\prime }\right)}^2+36xy{y}^{\prime }+6y^2=0$$

$$x^5{y}^{\prime \prime }+(2x^4-x)y^{\prime }-(2x^3-1)y=0$$

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

$$x^2{y}^{\prime \prime \prime }-5x{y}^{\prime \prime }+(4x^4+5)y^{\prime }-8x^{3}y=0$$

$$y^{\prime \prime }+2\cot (x)y^{\prime }+2\tan (x){\left(y^{\prime }\right)}^2=0$$

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

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

$$y^{\prime \prime }y-{\left(y^{\prime }\right)}^2=y^2\ln (y)-x^2{y}^2$$

$$({\sin }^2(x))y^{\prime \prime }-2y=0$$