ODE
\[ y'(x) \left (a+x \sqrt {y'(x)^2+1}\right )=y(x) \sqrt {y'(x)^2+1} \] ODE Classification
[_Clairaut]
Book solution method
Clairaut’s equation and related types, \(f(y-x y', y')=0\)
Mathematica ✗
cpu = 600.575 (sec), leaf count = 0 , timed out
$Aborted
Maple ✓
cpu = 0.029 (sec), leaf count = 18
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,x+{{\it \_C1}\,a{\frac {1}{\sqrt {{{\it \_C1}}^{2}+1}}}} \right \} \] Mathematica raw input
DSolve[y'[x]*(a + x*Sqrt[1 + y'[x]^2]) == y[x]*Sqrt[1 + y'[x]^2],y[x],x]
Mathematica raw output
$Aborted
Maple raw input
dsolve(diff(y(x),x)*(a+x*(1+diff(y(x),x)^2)^(1/2)) = y(x)*(1+diff(y(x),x)^2)^(1/2), y(x),'implicit')
Maple raw output
y(x) = _C1*x+_C1*a/(_C1^2+1)^(1/2)