ODE
\[ \sqrt {a c-b^2} \left (x y'(x)-y(x)\right )+\sqrt {a+2 b y'(x)+c y'(x)^2}=0 \] ODE Classification
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut]
Book solution method
Clairaut’s equation and related types, main form
Mathematica ✗
cpu = 600.203 (sec), leaf count = 0 , timed out
$Aborted
Maple ✓
cpu = 0.209 (sec), leaf count = 178
\[ \left \{ y \left ( x \right ) +{\frac {bx}{c}}-{a{\frac {1}{\sqrt {ca-{b}^{2}}}}{\frac {1}{\sqrt {-{\frac {ca}{a{x}^{2}c-{b}^{2}{x}^{2}-c}}+{\frac {{b}^{2}}{a{x}^{2}c-{b}^{2}{x}^{2}-c}}}}}}+{\frac {{b}^{2}}{c}{\frac {1}{\sqrt {ca-{b}^{2}}}}{\frac {1}{\sqrt {-{\frac {ca}{a{x}^{2}c-{b}^{2}{x}^{2}-c}}+{\frac {{b}^{2}}{a{x}^{2}c-{b}^{2}{x}^{2}-c}}}}}}=0,y \left ( x \right ) ={\it \_C1}\,x+{1\sqrt {c{{\it \_C1}}^{2}+2\,{\it \_C1}\,b+a}{\frac {1}{\sqrt {ca-{b}^{2}}}}} \right \} \] Mathematica raw input
DSolve[Sqrt[-b^2 + a*c]*(-y[x] + x*y'[x]) + Sqrt[a + 2*b*y'[x] + c*y'[x]^2] == 0,y[x],x]
Mathematica raw output
$Aborted
Maple raw input
dsolve((a+2*b*diff(y(x),x)+c*diff(y(x),x)^2)^(1/2)+(x*diff(y(x),x)-y(x))*(a*c-b^2)^(1/2) = 0, y(x),'implicit')
Maple raw output
y(x)+1/c*b*x-1/(a*c-b^2)^(1/2)/(-1/(a*c*x^2-b^2*x^2-c)*c*a+1/(a*c*x^2-b^2*x^2-c)*b^2)^(1/2)*a+1/(a*c-b^2)^(1/2)/(-1/(a*c*x^2-b^2*x^2-c)*c*a+1/(a*c*x^2-b^2*x^2-c)*b^2)^(1/2)/c*b^2 = 0, y(x) = _C1*x+(_C1^2*c+2*_C1*b+a)^(1/2)/(a*c-b^2)^(1/2)