ODE
\[ \sqrt {a^2+x^2} \left (b y'(x)^2+y(x) y''(x)\right )=y(x) y'(x) \] ODE Classification
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 2.23204 (sec), leaf count = 124
\[\left \{\left \{y(x)\to c_2 \exp \left (\int _1^x\frac {\sqrt {\frac {K[2]}{\sqrt {a^2+K[2]^2}}+1}}{\sqrt {1-\frac {K[2]}{\sqrt {a^2+K[2]^2}}} \left (c_1-\int _1^{K[2]}-\frac {(b+1) \sqrt {\frac {K[1]}{\sqrt {a^2+K[1]^2}}+1}}{\sqrt {1-\frac {K[1]}{\sqrt {a^2+K[1]^2}}}}dK[1]\right )}dK[2]\right )\right \}\right \}\]
Maple ✓
cpu = 0.529 (sec), leaf count = 68
\[\left [y \left (x \right ) = 2^{-\frac {1}{b +1}} \left (\frac {1}{\left (b +1\right ) \left (\textit {\_C1} \,a^{2} \ln \left (x +\sqrt {a^{2}+x^{2}}\right )+\textit {\_C1} x \sqrt {a^{2}+x^{2}}+x^{2} \textit {\_C1} +2 \textit {\_C2} \right )}\right )^{-\frac {1}{b +1}}\right ]\] Mathematica raw input
DSolve[Sqrt[a^2 + x^2]*(b*y'[x]^2 + y[x]*y''[x]) == y[x]*y'[x],y[x],x]
Mathematica raw output
{{y[x] -> E^Inactive[Integrate][Sqrt[1 + K[2]/Sqrt[a^2 + K[2]^2]]/(Sqrt[1 - K[2]/Sqrt[a^2 + K[2]^2]]*(C[1] - Inactive[Integrate][-(((1 + b)*Sqrt[1 + K[1]/Sqrt[a^2 + K[1]^2]])/Sqrt[1 - K[1]/Sqrt[a^2 + K[1]^2]]), {K[1], 1, K[2]}])), {K[2], 1, x}]*C[2]}}
Maple raw input
dsolve((a^2+x^2)^(1/2)*(y(x)*diff(diff(y(x),x),x)+b*diff(y(x),x)^2) = y(x)*diff(y(x),x), y(x))
Maple raw output
[y(x) = 1/(2^(1/(b+1)))/((1/(b+1)/(_C1*a^2*ln(x+(a^2+x^2)^(1/2))+_C1*x*(a^2+x^2)^(1/2)+x^2*_C1+2*_C2))^(1/(b+1)))]