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 = 4.3849 (sec), leaf count = 54
\[\text {Solve}\left [\left \{x=\frac {\sqrt {K[1]^2+1}-\sinh ^{-1}(K[1])+c_1}{(K[1]-1)^2},\sqrt {K[1]^2+1}+y(x)=x K[1]^2\right \},\{y(x),K[1]\}\right ]\]
Maple ✓
cpu = 0.7 (sec), leaf count = 583
\[\left [\frac {x^{2} \textit {\_C1}}{\left (\sqrt {4 x y \left (x \right )+2+2 \sqrt {4 x^{2}+4 x y \left (x \right )+1}}-2 x \right )^{2}}+x -\frac {2 x^{2} \left (\sqrt {2}\, \sqrt {\frac {2 x^{2}+2 x y \left (x \right )+\sqrt {4 x^{2}+4 x y \left (x \right )+1}+1}{x^{2}}}-2 \arcsinh \left (\frac {\sqrt {4 x y \left (x \right )+2+2 \sqrt {4 x^{2}+4 x y \left (x \right )+1}}}{2 x}\right )\right )}{\left (\sqrt {4 x y \left (x \right )+2+2 \sqrt {4 x^{2}+4 x y \left (x \right )+1}}-2 x \right )^{2}} = 0, \frac {x^{2} \textit {\_C1}}{\left (\sqrt {4 x y \left (x \right )+2+2 \sqrt {4 x^{2}+4 x y \left (x \right )+1}}+2 x \right )^{2}}+x -\frac {2 x^{2} \left (\sqrt {2}\, \sqrt {\frac {2 x^{2}+2 x y \left (x \right )+\sqrt {4 x^{2}+4 x y \left (x \right )+1}+1}{x^{2}}}+2 \arcsinh \left (\frac {\sqrt {4 x y \left (x \right )+2+2 \sqrt {4 x^{2}+4 x y \left (x \right )+1}}}{2 x}\right )\right )}{\left (\sqrt {4 x y \left (x \right )+2+2 \sqrt {4 x^{2}+4 x y \left (x \right )+1}}+2 x \right )^{2}} = 0, \frac {x^{2} \textit {\_C1}}{\left (\sqrt {4 x y \left (x \right )-2 \sqrt {4 x^{2}+4 x y \left (x \right )+1}+2}-2 x \right )^{2}}+x +\frac {2 x^{2} \left (-\sqrt {\frac {4 x^{2}+4 x y \left (x \right )-2 \sqrt {4 x^{2}+4 x y \left (x \right )+1}+2}{x^{2}}}+2 \arcsinh \left (\frac {\sqrt {4 x y \left (x \right )-2 \sqrt {4 x^{2}+4 x y \left (x \right )+1}+2}}{2 x}\right )\right )}{\left (\sqrt {4 x y \left (x \right )-2 \sqrt {4 x^{2}+4 x y \left (x \right )+1}+2}-2 x \right )^{2}} = 0, \frac {x^{2} \textit {\_C1}}{\left (\sqrt {4 x y \left (x \right )-2 \sqrt {4 x^{2}+4 x y \left (x \right )+1}+2}+2 x \right )^{2}}+x -\frac {2 x^{2} \left (\sqrt {\frac {4 x^{2}+4 x y \left (x \right )-2 \sqrt {4 x^{2}+4 x y \left (x \right )+1}+2}{x^{2}}}+2 \arcsinh \left (\frac {\sqrt {4 x y \left (x \right )-2 \sqrt {4 x^{2}+4 x y \left (x \right )+1}+2}}{2 x}\right )\right )}{\left (\sqrt {4 x y \left (x \right )-2 \sqrt {4 x^{2}+4 x y \left (x \right )+1}+2}+2 x \right )^{2}} = 0\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 == (-ArcSinh[K[1]] + C[1] + Sqrt[1 + K[1]^2])/(-1 + K[1])^2, Sqrt[1 + K[1]^2] + y[x] == x*K[1]^2}, {y[x], K[1]}]
Maple raw input
dsolve((1+diff(y(x),x)^2)^(1/2)-x*diff(y(x),x)^2+y(x) = 0, y(x))
Maple raw output
[x^2/((4*x*y(x)+2+2*(4*x^2+4*x*y(x)+1)^(1/2))^(1/2)-2*x)^2*_C1+x-2*x^2*(2^(1/2)*((2*x^2+2*x*y(x)+(4*x^2+4*x*y(x)+1)^(1/2)+1)/x^2)^(1/2)-2*arcsinh(1/2/x*(4*x*y(x)+2+2*(4*x^2+4*x*y(x)+1)^(1/2))^(1/2)))/((4*x*y(x)+2+2*(4*x^2+4*x*y(x)+1)^(1/2))^(1/2)-2*x)^2 = 0, x^2/((4*x*y(x)+2+2*(4*x^2+4*x*y(x)+1)^(1/2))^(1/2)+2*x)^2*_C1+x-2*x^2*(2^(1/2)*((2*x^2+2*x*y(x)+(4*x^2+4*x*y(x)+1)^(1/2)+1)/x^2)^(1/2)+2*arcsinh(1/2/x*(4*x*y(x)+2+2*(4*x^2+4*x*y(x)+1)^(1/2))^(1/2)))/((4*x*y(x)+2+2*(4*x^2+4*x*y(x)+1)^(1/2))^(1/2)+2*x)^2 = 0, x^2/((4*x*y(x)-2*(4*x^2+4*x*y(x)+1)^(1/2)+2)^(1/2)-2*x)^2*_C1+x+2*x^2*(-((4*x^2+4*x*y(x)-2*(4*x^2+4*x*y(x)+1)^(1/2)+2)/x^2)^(1/2)+2*arcsinh(1/2/x*(4*x*y(x)-2*(4*x^2+4*x*y(x)+1)^(1/2)+2)^(1/2)))/((4*x*y(x)-2*(4*x^2+4*x*y(x)+1)^(1/2)+2)^(1/2)-2*x)^2 = 0, x^2/((4*x*y(x)-2*(4*x^2+4*x*y(x)+1)^(1/2)+2)^(1/2)+2*x)^2*_C1+x-2*x^2*(((4*x^2+4*x*y(x)-2*(4*x^2+4*x*y(x)+1)^(1/2)+2)/x^2)^(1/2)+2*arcsinh(1/2/x*(4*x*y(x)-2*(4*x^2+4*x*y(x)+1)^(1/2)+2)^(1/2)))/((4*x*y(x)-2*(4*x^2+4*x*y(x)+1)^(1/2)+2)^(1/2)+2*x)^2 = 0]