ODE
\[ y'(x) \log \left (\sqrt {a+y'(x)^2}+y'(x)\right )-x y'(x)-\sqrt {y'(x)^2+1}+y(x)=0 \] ODE Classification
[_Clairaut]
Book solution method
Clairaut’s equation and related types, \(f(y-x y', y')=0\)
Mathematica ✓
cpu = 0.157658 (sec), leaf count = 38
\[\left \{\left \{y(x)\to -c_1 \log \left (\sqrt {a+c_1{}^2}+c_1\right )+c_1 x+\sqrt {1+c_1{}^2}\right \}\right \}\]
Maple ✓
cpu = 8.651 (sec), leaf count = 129
\[\left [\left [x \left (\textit {\_T} \right ) = \frac {\ln \left (\textit {\_T} +\sqrt {\textit {\_T}^{2}+a}\right ) \sqrt {\textit {\_T}^{2}+a}\, \sqrt {\textit {\_T}^{2}+1}+\sqrt {\textit {\_T}^{2}+1}\, \textit {\_T} -\textit {\_T} \sqrt {\textit {\_T}^{2}+a}}{\sqrt {\textit {\_T}^{2}+a}\, \sqrt {\textit {\_T}^{2}+1}}, y \left (\textit {\_T} \right ) = \frac {\sqrt {\textit {\_T}^{2}+1}\, \textit {\_T}^{2}+\sqrt {\textit {\_T}^{2}+a}}{\sqrt {\textit {\_T}^{2}+1}\, \sqrt {\textit {\_T}^{2}+a}}\right ], y \left (x \right ) = -\textit {\_C1} \ln \left (\textit {\_C1} +\sqrt {\textit {\_C1}^{2}+a}\right )+\sqrt {\textit {\_C1}^{2}+1}+\textit {\_C1} x\right ]\] Mathematica raw input
DSolve[y[x] - x*y'[x] + Log[y'[x] + Sqrt[a + y'[x]^2]]*y'[x] - Sqrt[1 + y'[x]^2] == 0,y[x],x]
Mathematica raw output
{{y[x] -> x*C[1] + Sqrt[1 + C[1]^2] - C[1]*Log[C[1] + Sqrt[a + C[1]^2]]}}
Maple raw input
dsolve(diff(y(x),x)*ln(diff(y(x),x)+(a+diff(y(x),x)^2)^(1/2))-(1+diff(y(x),x)^2)^(1/2)-x*diff(y(x),x)+y(x) = 0, y(x))
Maple raw output
[[x(_T) = (ln(_T+(_T^2+a)^(1/2))*(_T^2+a)^(1/2)*(_T^2+1)^(1/2)+(_T^2+1)^(1/2)*_T-_T*(_T^2+a)^(1/2))/(_T^2+a)^(1/2)/(_T^2+1)^(1/2), y(_T) = ((_T^2+1)^(1/2)*_T^2+(_T^2+a)^(1/2))/(_T^2+1)^(1/2)/(_T^2+a)^(1/2)], y(x) = -_C1*ln(_C1+(_C1^2+a)^(1/2))+(_C1^2+1)^(1/2)+_C1*x]