ODE
\[ (a-1) b+a x^2+2 a x y(x) y'(x)+(1-a) y(x)^2+y(x)^2 y'(x)^2=0 \] ODE Classification
[_rational, [_1st_order, `_with_symmetry_[F(x),G(y)]`]]
Book solution method
Homogeneous ODE, \(x^n f\left ( \frac {y}{x} , y' \right )=0\), Solve for \(x\)
Mathematica ✓
cpu = 0.955333 (sec), leaf count = 65
\[\left \{\left \{y(x)\to -\sqrt {-2 (a-1) c_1 x+(a-1) c_1^2+b-x^2}\right \},\left \{y(x)\to \sqrt {-2 (a-1) c_1 x+(a-1) c_1^2+b-x^2}\right \}\right \}\]
Maple ✓
cpu = 1.638 (sec), leaf count = 126
\[ \left \{ \left ( y \left ( x \right ) \right ) ^{2}+a{x}^{2}-b=0,[x \left ( {\it \_T} \right ) ={\frac {1}{a} \left ( -{\it \_C1}\,{\it \_T}\,a+\sqrt { \left ( {{\it \_C1}}^{2}-b \right ) \left ( a-1 \right ) a}\sqrt {{{\it \_T}}^{2}+1} \right ) {\frac {1}{\sqrt {{{\it \_T}}^{2}+1}}}},y \left ( {\it \_T} \right ) ={{\it \_C1}{\frac {1}{\sqrt {{{\it \_T}}^{2}+1}}}}],[x \left ( {\it \_T} \right ) =-{\frac {1}{a} \left ( {\it \_C1}\,{\it \_T}\,a+\sqrt { \left ( {{\it \_C1}}^{2}-b \right ) \left ( a-1 \right ) a}\sqrt {{{\it \_T}}^{2}+1} \right ) {\frac {1}{\sqrt {{{\it \_T}}^{2}+1}}}},y \left ( {\it \_T} \right ) ={{\it \_C1}{\frac {1}{\sqrt {{{\it \_T}}^{2}+1}}}}] \right \} \] Mathematica raw input
DSolve[(-1 + a)*b + a*x^2 + (1 - a)*y[x]^2 + 2*a*x*y[x]*y'[x] + y[x]^2*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> -Sqrt[b - x^2 - 2*(-1 + a)*x*C[1] + (-1 + a)*C[1]^2]}, {y[x] -> Sqrt[b - x^2 - 2*(-1 + a)*x*C[1] + (-1 + a)*C[1]^2]}}
Maple raw input
dsolve(y(x)^2*diff(y(x),x)^2+2*a*x*y(x)*diff(y(x),x)+(a-1)*b+a*x^2+(1-a)*y(x)^2 = 0, y(x),'implicit')
Maple raw output
y(x)^2+a*x^2-b = 0, [x(_T) = (-_C1*_T*a+((_C1^2-b)*(a-1)*a)^(1/2)*(_T^2+1)^(1/2))/(_T^2+1)^(1/2)/a, y(_T) = _C1/(_T^2+1)^(1/2)], [x(_T) = -(_C1*_T*a+((_C1^2-b)*(a-1)*a)^(1/2)*(_T^2+1)^(1/2))/(_T^2+1)^(1/2)/a, y(_T) = _C1/(_T^2+1)^(1/2)]