ODE
\[ -x y'(x)^2+\sqrt {y'(x)^2+1}+y(x)=0 \] 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
\[\text {Solve}\left [\left \{x=\frac {c_1+\sqrt {\text {K$\$$258298}^2+1}-\sinh ^{-1}(\text {K$\$$258298})}{(\text {K$\$$258298}-1)^2},\sqrt {\text {K$\$$258298}^2+1}+y(x)=\text {K$\$$258298}^2 x\right \},\{y(x),\text {K$\$$258298}\}\right ]\]
Maple ✓
cpu = 0.033 (sec), leaf count = 57
\[ \left \{ [x \left ( {\it \_T} \right ) ={\frac {1}{ \left ( {\it \_T}-1 \right ) ^{2}} \left ( \sqrt {{{\it \_T}}^{2}+1}-{\it Arcsinh} \left ( {\it \_T} \right ) +{\it \_C1} \right ) },y \left ( {\it \_T} \right ) ={\frac {1}{ \left ( {\it \_T}-1 \right ) ^{2}} \left ( \left ( 2\,{\it \_T}-1 \right ) \sqrt {{{\it \_T}}^{2}+1}+{{\it \_T}}^{2} \left ( -{\it Arcsinh} \left ( {\it \_T} \right ) +{\it \_C1} \right ) \right ) }] \right \} \] 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]