ODE
\[ a \sqrt {y'(x)^2+1}+x y'(x)-y(x)=0 \] ODE Classification
[[_1st_order, _with_linear_symmetries], _Clairaut]
Book solution method
No Missing Variables ODE, Solve for \(y'\)
Mathematica ✓
cpu = 0.005442 (sec), leaf count = 22
\[\left \{\left \{y(x)\to a \sqrt {c_1^2+1}+c_1 x\right \}\right \}\]
Maple ✓
cpu = 0.028 (sec), leaf count = 17
\[ \left \{ y \left ( x \right ) =a\sqrt {{{\it \_C1}}^{2}+1}+{\it \_C1}\,x \right \} \] Mathematica raw input
DSolve[-y[x] + x*y'[x] + a*Sqrt[1 + y'[x]^2] == 0,y[x],x]
Mathematica raw output
{{y[x] -> x*C[1] + a*Sqrt[1 + C[1]^2]}}
Maple raw input
dsolve(a*(1+diff(y(x),x)^2)^(1/2)+x*diff(y(x),x)-y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = a*(_C1^2+1)^(1/2)+_C1*x