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 = 0.0740249 (sec), leaf count = 38
\[\left \{\left \{y(x)\to -c_1 \log \left (\sqrt {a+c_1^2}+c_1\right )+c_1 x+\sqrt {c_1^2+1}\right \}\right \}\]
Maple ✓
cpu = 2.413 (sec), leaf count = 129
\[ \left \{ y \left ( x \right ) =-{\it \_C1}\,\ln \left ( {\it \_C1}+\sqrt {{{\it \_C1}}^{2}+a} \right ) +\sqrt {{{\it \_C1}}^{2}+1}+{\it \_C1}\,x,[x \left ( {\it \_T} \right ) ={1 \left ( \ln \left ( {\it \_T}+\sqrt {{{\it \_T}}^{2}+a} \right ) \sqrt {{{\it \_T}}^{2}+a}\sqrt {{{\it \_T}}^{2}+1}+{\it \_T}\, \left ( \sqrt {{{\it \_T}}^{2}+1}-\sqrt {{{\it \_T}}^{2}+a} \right ) \right ) {\frac {1}{\sqrt {{{\it \_T}}^{2}+a}}}{\frac {1}{\sqrt {{{\it \_T}}^{2}+1}}}},y \left ( {\it \_T} \right ) ={1 \left ( \sqrt {{{\it \_T}}^{2}+1}{{\it \_T}}^{2}+\sqrt {{{\it \_T}}^{2}+a} \right ) {\frac {1}{\sqrt {{{\it \_T}}^{2}+a}}}{\frac {1}{\sqrt {{{\it \_T}}^{2}+1}}}}] \right \} \] 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),'implicit')
Maple raw output
[x(_T) = 1/(_T^2+a)^(1/2)/(_T^2+1)^(1/2)*(ln(_T+(_T^2+a)^(1/2))*(_T^2+a)^(1/2)*(_T^2+1)^(1/2)+_T*((_T^2+1)^(1/2)-(_T^2+a)^(1/2))), y(_T) = ((_T^2+1)^(1/2)*_T^2+(_T^2+a)^(1/2))/(_T^2+a)^(1/2)/(_T^2+1)^(1/2)], y(x) = -_C1*ln(_C1+(_C1^2+a)^(1/2))+(_C1^2+1)^(1/2)+_C1*x