ODE
\[ -(a+2 x y(x)) y'(x)+x^2 y'(x)^2+y(x)^2=0 \] ODE Classification
[[_homogeneous, `class G`], _rational, _Clairaut]
Book solution method
Clairaut’s equation and related types, \(f(y-x y', y')=0\)
Mathematica ✓
cpu = 0.273771 (sec), leaf count = 47
\[\left \{\left \{y(x)\to \frac {x-2 \sqrt {a} c_1}{4 c_1^2}\right \},\left \{y(x)\to \frac {2 \sqrt {a} c_1+x}{4 c_1^2}\right \}\right \}\]
Maple ✓
cpu = 0.044 (sec), leaf count = 36
\[ \left \{ y \left ( x \right ) =-{\frac {a}{4\,x}},y \left ( x \right ) ={\it \_C1}\,x-\sqrt {a{\it \_C1}},y \left ( x \right ) ={\it \_C1}\,x+\sqrt {a{\it \_C1}} \right \} \] Mathematica raw input
DSolve[y[x]^2 - (a + 2*x*y[x])*y'[x] + x^2*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> (x - 2*Sqrt[a]*C[1])/(4*C[1]^2)}, {y[x] -> (x + 2*Sqrt[a]*C[1])/(4*C[1]^2)}}
Maple raw input
dsolve(x^2*diff(y(x),x)^2-(a+2*x*y(x))*diff(y(x),x)+y(x)^2 = 0, y(x),'implicit')
Maple raw output
y(x) = -1/4/x*a, y(x) = _C1*x-(a*_C1)^(1/2), y(x) = _C1*x+(a*_C1)^(1/2)