ODE
\[ -a y(x) y'(x)-a x+\sqrt {y'(x)^2+1}=0 \] ODE Classification
[[_1st_order, _with_linear_symmetries], _dAlembert]
Book solution method
No Missing Variables ODE, Solve for \(x\)
Mathematica ✓
cpu = 1.34863 (sec), leaf count = 105
\[\left \{\text {Solve}\left [\tan ^{-1}\left (\sqrt {a^2 \left (x^2+y(x)^2\right )-1}\right )+\tan ^{-1}\left (\frac {x}{y(x)}\right )=\sqrt {a^2 \left (x^2+y(x)^2\right )-1}+c_1,y(x)\right ],\text {Solve}\left [\sqrt {a^2 \left (x^2+y(x)^2\right )-1}+\tan ^{-1}\left (\frac {x}{y(x)}\right )=\tan ^{-1}\left (\sqrt {a^2 \left (x^2+y(x)^2\right )-1}\right )+c_1,y(x)\right ]\right \}\]
Maple ✓
cpu = 3.698 (sec), leaf count = 357
\[\left [y \left (x \right ) = -\frac {a x -\sqrt {\tan ^{2}\left (\RootOf \left (4 \sin \left (\textit {\_Z} \right ) \textit {\_C1} \,a^{2} x +a^{2} \textit {\_C1}^{2} \cos \left (2 \textit {\_Z} \right )+4 \sin \left (\textit {\_Z} \right ) a x \textit {\_Z} +2 \textit {\_C1} \textit {\_Z} a \cos \left (2 \textit {\_Z} \right )-a^{2} \textit {\_C1}^{2}-2 a^{2} x^{2}+\textit {\_Z}^{2} \cos \left (2 \textit {\_Z} \right )-2 \textit {\_C1} \textit {\_Z} a -\textit {\_Z}^{2}+\cos \left (2 \textit {\_Z} \right )+1\right )\right )+1}}{a \tan \left (\RootOf \left (4 \sin \left (\textit {\_Z} \right ) \textit {\_C1} \,a^{2} x +a^{2} \textit {\_C1}^{2} \cos \left (2 \textit {\_Z} \right )+4 \sin \left (\textit {\_Z} \right ) a x \textit {\_Z} +2 \textit {\_C1} \textit {\_Z} a \cos \left (2 \textit {\_Z} \right )-a^{2} \textit {\_C1}^{2}-2 a^{2} x^{2}+\textit {\_Z}^{2} \cos \left (2 \textit {\_Z} \right )-2 \textit {\_C1} \textit {\_Z} a -\textit {\_Z}^{2}+\cos \left (2 \textit {\_Z} \right )+1\right )\right )}, y \left (x \right ) = -\frac {a x -\sqrt {\tan ^{2}\left (\RootOf \left (-4 \sin \left (\textit {\_Z} \right ) \textit {\_C1} \,a^{2} x +a^{2} \textit {\_C1}^{2} \cos \left (2 \textit {\_Z} \right )-4 \sin \left (\textit {\_Z} \right ) a x \textit {\_Z} +2 \textit {\_C1} \textit {\_Z} a \cos \left (2 \textit {\_Z} \right )-a^{2} \textit {\_C1}^{2}-2 a^{2} x^{2}+\textit {\_Z}^{2} \cos \left (2 \textit {\_Z} \right )-2 \textit {\_C1} \textit {\_Z} a -\textit {\_Z}^{2}+\cos \left (2 \textit {\_Z} \right )+1\right )\right )+1}}{a \tan \left (\RootOf \left (-4 \sin \left (\textit {\_Z} \right ) \textit {\_C1} \,a^{2} x +a^{2} \textit {\_C1}^{2} \cos \left (2 \textit {\_Z} \right )-4 \sin \left (\textit {\_Z} \right ) a x \textit {\_Z} +2 \textit {\_C1} \textit {\_Z} a \cos \left (2 \textit {\_Z} \right )-a^{2} \textit {\_C1}^{2}-2 a^{2} x^{2}+\textit {\_Z}^{2} \cos \left (2 \textit {\_Z} \right )-2 \textit {\_C1} \textit {\_Z} a -\textit {\_Z}^{2}+\cos \left (2 \textit {\_Z} \right )+1\right )\right )}\right ]\] Mathematica raw input
DSolve[-(a*x) - a*y[x]*y'[x] + Sqrt[1 + y'[x]^2] == 0,y[x],x]
Mathematica raw output
{Solve[ArcTan[x/y[x]] + ArcTan[Sqrt[-1 + a^2*(x^2 + y[x]^2)]] == C[1] + Sqrt[-1 + a^2*(x^2 + y[x]^2)], y[x]], Solve[ArcTan[x/y[x]] + Sqrt[-1 + a^2*(x^2 + y[x]^2)] == ArcTan[Sqrt[-1 + a^2*(x^2 + y[x]^2)]] + C[1], y[x]]}
Maple raw input
dsolve((1+diff(y(x),x)^2)^(1/2)-a*y(x)*diff(y(x),x)-a*x = 0, y(x))
Maple raw output
[y(x) = -(a*x-(tan(RootOf(4*sin(_Z)*_C1*a^2*x+a^2*_C1^2*cos(2*_Z)+4*sin(_Z)*a*x*_Z+2*_C1*_Z*a*cos(2*_Z)-a^2*_C1^2-2*a^2*x^2+_Z^2*cos(2*_Z)-2*_C1*_Z*a-_Z^2+cos(2*_Z)+1))^2+1)^(1/2))/a/tan(RootOf(4*sin(_Z)*_C1*a^2*x+a^2*_C1^2*cos(2*_Z)+4*sin(_Z)*a*x*_Z+2*_C1*_Z*a*cos(2*_Z)-a^2*_C1^2-2*a^2*x^2+_Z^2*cos(2*_Z)-2*_C1*_Z*a-_Z^2+cos(2*_Z)+1)), y(x) = -(a*x-(tan(RootOf(-4*sin(_Z)*_C1*a^2*x+a^2*_C1^2*cos(2*_Z)-4*sin(_Z)*a*x*_Z+2*_C1*_Z*a*cos(2*_Z)-a^2*_C1^2-2*a^2*x^2+_Z^2*cos(2*_Z)-2*_C1*_Z*a-_Z^2+cos(2*_Z)+1))^2+1)^(1/2))/a/tan(RootOf(-4*sin(_Z)*_C1*a^2*x+a^2*_C1^2*cos(2*_Z)-4*sin(_Z)*a*x*_Z+2*_C1*_Z*a*cos(2*_Z)-a^2*_C1^2-2*a^2*x^2+_Z^2*cos(2*_Z)-2*_C1*_Z*a-_Z^2+cos(2*_Z)+1))]