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 = 96.7453 (sec), leaf count = 407
\[\left \{\text {Solve}\left [\sqrt {a^2 \left (x^2+y(x)^2\right )-1}+\frac {1}{4} i \log \left (\left (a^2 x^2-1\right ) y(x)^2-2 x y(x) \sqrt {a^2 x^2+a^2 y(x)^2-1}+a^2 y(x)^4+x^2\right )+\frac {1}{4} i \log \left (\left (a^2 x^2-1\right ) y(x)^2+2 x y(x) \sqrt {a^2 x^2+a^2 y(x)^2-1}+a^2 y(x)^4+x^2\right )+\tan ^{-1}\left (\frac {1}{\sqrt {a^2 \left (x^2+y(x)^2\right )-1}}\right )-\frac {1}{2} i \log \left (1-a^2 y(x)^2\right )-\frac {1}{2} i \log \left (x^2+y(x)^2\right )+\tan ^{-1}\left (\frac {x}{y(x)}\right )=c_1,y(x)\right ],\text {Solve}\left [4 \sqrt {a^2 x^2+a^2 y(x)^2-1}+i \log \left (a^2 x^2 y(x)^2-2 x y(x) \sqrt {a^2 \left (x^2+y(x)^2\right )-1}+a^2 y(x)^4+x^2-y(x)^2\right )+i \log \left (a^2 x^2 y(x)^2+2 x y(x) \sqrt {a^2 \left (x^2+y(x)^2\right )-1}+a^2 y(x)^4+x^2-y(x)^2\right )+4 \tan ^{-1}\left (\frac {1}{\sqrt {a^2 \left (x^2+y(x)^2\right )-1}}\right )-2 i \log \left (a^2 y(x)^2-1\right )+4 c_1-2 i \log \left (x^2+y(x)^2\right )=4 \tan ^{-1}\left (\frac {x}{y(x)}\right ),y(x)\right ]\right \}\]
Maple ✓
cpu = 0.043 (sec), leaf count = 53
\[ \left \{ [x \left ( {\it \_T} \right ) ={{\it \_T} \left ( {\frac {\arctan \left ( {\it \_T} \right ) }{a}}+{\frac {1}{{\it \_T}\,a}}+{\it \_C1} \right ) {\frac {1}{\sqrt {{{\it \_T}}^{2}+1}}}},y \left ( {\it \_T} \right ) =-{\frac {{\it \_C1}\,a-{\it \_T}+\arctan \left ( {\it \_T} \right ) }{a}{\frac {1}{\sqrt {{{\it \_T}}^{2}+1}}}}] \right \} \] 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[1/Sqrt[-1 + a^2*(x^2 + y[x]^2)]] - (I/2)*Log[x^2 + y[x]^2] - (I/2)*Log[1 - a^2*y[x]^2] + (I/4)*Log[x^2 + (-1 + a^2*x^2)*y[x]^2 + a^2*y[x]^4 - 2*x*y[x]*Sqrt[-1 + a^2*x^2 + a^2*y[x]^2]] + (I/4)*Log[x^2 + (-1 + a^2*x^2)*y[x]^2 + a^2*y[x]^4 + 2*x*y[x]*Sqrt[-1 + a^2*x^2 + a^2*y[x]^2]] + Sqrt[-1 + a^2*(x^2 + y[x]^2)] == C[1], y[x]], Solve[4*ArcTan[1/Sqrt[-1 + a^2*(x^2 + y[x]^2)]] + 4*C[1] - (2*I)*Log[x^2 + y[x]^2] - (2*I)*Log[-1 + a^2*y[x]^2] + I*Log[x^2 - y[x]^2 + a^2*x^2*y[x]^2 + a^2*y[x]^4 - 2*x*y[x]*Sqrt[-1 + a^2*(x^2 + y[x]^2)]] + I*Log[x^2 - y[x]^2 + a^2*x^2*y[x]^2 + a^2*y[x]^4 + 2*x*y[x]*Sqrt[-1 + a^2*(x^2 + y[x]^2)]] + 4*Sqrt[-1 + a^2*x^2 + a^2*y[x]^2] == 4*ArcTan[x/y[x]], 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),'implicit')
Maple raw output
[x(_T) = 1/(_T^2+1)^(1/2)*_T*(1/a*arctan(_T)+1/_T/a+_C1), y(_T) = -(_C1*a-_T+arctan(_T))/(_T^2+1)^(1/2)/a]