ODE
\[ a-x^2-2 x y(x) y'(x)+y(x)^2 y'(x)^2+2 y(x)^2=0 \] ODE Classification
[_rational, [_1st_order, `_with_symmetry_[F(x),G(y)]`]]
Book solution method
Change of variable
Mathematica ✓
cpu = 0.570202 (sec), leaf count = 63
\[\left \{\left \{y(x)\to -\sqrt {-\frac {a}{2}+4 c_1 x-2 c_1^2-x^2}\right \},\left \{y(x)\to \sqrt {-\frac {a}{2}+4 c_1 x-2 c_1^2-x^2}\right \}\right \}\]
Maple ✓
cpu = 0.41 (sec), leaf count = 108
\[ \left \{ \left ( y \left ( x \right ) \right ) ^{2}-{x}^{2}+{\frac {a}{2}}=0,[x \left ( {\it \_T} \right ) ={1 \left ( \sqrt {2\,{{\it \_C1}}^{2}+a}\sqrt {{{\it \_T}}^{2}+1}-{\it \_C1}\,{\it \_T} \right ) {\frac {1}{\sqrt {{{\it \_T}}^{2}+1}}}},y \left ( {\it \_T} \right ) ={{\it \_C1}{\frac {1}{\sqrt {{{\it \_T}}^{2}+1}}}}],[x \left ( {\it \_T} \right ) =-{1 \left ( \sqrt {2\,{{\it \_C1}}^{2}+a}\sqrt {{{\it \_T}}^{2}+1}+{\it \_C1}\,{\it \_T} \right ) {\frac {1}{\sqrt {{{\it \_T}}^{2}+1}}}},y \left ( {\it \_T} \right ) ={{\it \_C1}{\frac {1}{\sqrt {{{\it \_T}}^{2}+1}}}}] \right \} \] Mathematica raw input
DSolve[a - x^2 + 2*y[x]^2 - 2*x*y[x]*y'[x] + y[x]^2*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> -Sqrt[-a/2 - x^2 + 4*x*C[1] - 2*C[1]^2]}, {y[x] -> Sqrt[-a/2 - x^2 + 4*x*C[1] - 2*C[1]^2]}}
Maple raw input
dsolve(y(x)^2*diff(y(x),x)^2-2*x*y(x)*diff(y(x),x)+a-x^2+2*y(x)^2 = 0, y(x),'implicit')
Maple raw output
y(x)^2-x^2+1/2*a = 0, [x(_T) = -((2*_C1^2+a)^(1/2)*(_T^2+1)^(1/2)+_C1*_T)/(_T^2+1)^(1/2), y(_T) = _C1/(_T^2+1)^(1/2)], [x(_T) = ((2*_C1^2+a)^(1/2)*(_T^2+1)^(1/2)-_C1*_T)/(_T^2+1)^(1/2), y(_T) = _C1/(_T^2+1)^(1/2)]