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 = 0.157008 (sec), leaf count = 22
\[\left \{\left \{y(x)\to c_1 \left (x+\frac {a}{\sqrt {1+c_1{}^2}}\right )\right \}\right \}\]
Maple ✓
cpu = 0.064 (sec), leaf count = 18
\[\left [y \left (x \right ) = \textit {\_C1} x +\frac {\textit {\_C1} a}{\sqrt {\textit {\_C1}^{2}+1}}\right ]\] 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
{{y[x] -> C[1]*(x + a/Sqrt[1 + C[1]^2])}}
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))
Maple raw output
[y(x) = _C1*x+_C1*a/(_C1^2+1)^(1/2)]