4.23.31 $-x{y}^{\prime }(x{)}^2+\sqrt{y^{\prime }(x{)}^2+1}+y(x)=0$

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

[_dAlembert]

Book solution method
Clairaut’s equation and related types, d’Alembert’s equation (also call Lagrange’s)

Mathematica ✓
cpu = 5.75295 (sec), leaf count = 48

$$\text{Solve}[\{x=\frac{c_1+\sqrt{{\text{K}\mathrm{\$}\text{258298}}^2+1}-{\sinh }^{-1}(\text{K}\mathrm{\$}\text{258298})}{(\text{K}\mathrm{\$}\text{258298}-1{)}^2},\sqrt{{\text{K}\mathrm{\$}\text{258298}}^2+1}+y(x)={\text{K}\mathrm{\$}\text{258298}}^{2}x\},\{y(x),\text{K}\mathrm{\$}\text{258298}\}]$$

Maple ✓
cpu = 0.033 (sec), leaf count = 57

$$\{[x\left(\mathit{\_}\mathit{T}\right)=\frac{1}{{(\mathit{\_}\mathit{T}-1)}^2}(\sqrt{{\mathit{\_}\mathit{T}}^2+1}-\mathit{A}\mathit{r}\mathit{c}\mathit{s}\mathit{i}\mathit{n}\mathit{h}\left(\mathit{\_}\mathit{T}\right)+\mathit{\_}\mathit{C}\mathit{1}),y\left(\mathit{\_}\mathit{T}\right)=\frac{1}{{(\mathit{\_}\mathit{T}-1)}^2}((2\mathit{\_}\mathit{T}-1)\sqrt{{\mathit{\_}\mathit{T}}^2+1}+{\mathit{\_}\mathit{T}}^2(-\mathit{A}\mathit{r}\mathit{c}\mathit{s}\mathit{i}\mathit{n}\mathit{h}\left(\mathit{\_}\mathit{T}\right)+\mathit{\_}\mathit{C}\mathit{1}))]\}$$ Mathematica raw input

DSolve[y[x] - x*y'[x]^2 + Sqrt[1 + y'[x]^2] == 0,y[x],x]

Mathematica raw output

Solve[{x == (Sqrt[1 + K$258298^2] - ArcSinh[K$258298] + C[1])/(-1 + K$258298)^2,
 Sqrt[1 + K$258298^2] + y[x] == K$258298^2*x}, {y[x], K$258298}]

Maple raw input

dsolve((1+diff(y(x),x)^2)^(1/2)-x*diff(y(x),x)^2+y(x) = 0, y(x),'implicit')

Maple raw output

[x(_T) = 1/(_T-1)^2*((_T^2+1)^(1/2)-arcsinh(_T)+_C1), y(_T) = ((2*_T-1)*(_T^2+1)
^(1/2)+_T^2*(-arcsinh(_T)+_C1))/(_T-1)^2]