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 = 0.167774 (sec), leaf count = 41
\[\left \{\left \{y(x)\to \frac {\sqrt {a+2 b c_1+c c_1{}^2}}{\sqrt {a c-b^2}}+c_1 x\right \}\right \}\]
Maple ✓
cpu = 0.324 (sec), leaf count = 178
\[\left [y \left (x \right ) = -\frac {b x}{c}+\frac {a}{\sqrt {c a -b^{2}}\, \sqrt {-\frac {c a}{a \,x^{2} c -b^{2} x^{2}-c}+\frac {b^{2}}{a \,x^{2} c -b^{2} x^{2}-c}}}-\frac {b^{2}}{\sqrt {c a -b^{2}}\, \sqrt {-\frac {c a}{a \,x^{2} c -b^{2} x^{2}-c}+\frac {b^{2}}{a \,x^{2} c -b^{2} x^{2}-c}}\, c}, y \left (x \right ) = \textit {\_C1} x +\frac {\sqrt {c \,\textit {\_C1}^{2}+2 \textit {\_C1} b +a}}{\sqrt {c a -b^{2}}}\right ]\] 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
{{y[x] -> x*C[1] + Sqrt[a + 2*b*C[1] + c*C[1]^2]/Sqrt[-b^2 + a*c]}}
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))
Maple raw output
[y(x) = -b/c*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, y(x) = _C1*x+(_C1^2*c+2*_C1*b+a)^(1/2)/(a*c-b^2)^(1/2)]