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 = 1.04534 (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 = 1.535 (sec), leaf count = 251
\[\left [y \left (x \right ) = \sqrt {-a \,x^{2}+b}, y \left (x \right ) = -\sqrt {-a \,x^{2}+b}, y \left (x \right ) = \frac {\sqrt {-a^{2} x^{2}-2 a \sqrt {\textit {\_C1} \,a^{2}-a^{2} b -\textit {\_C1} a +a b}\, x +\textit {\_C1} a +a^{2} b -a b}}{a}, y \left (x \right ) = \frac {\sqrt {-a^{2} x^{2}+2 a \sqrt {\textit {\_C1} \,a^{2}-a^{2} b -\textit {\_C1} a +a b}\, x +\textit {\_C1} a +a^{2} b -a b}}{a}, y \left (x \right ) = -\frac {\sqrt {-a^{2} x^{2}-2 a \sqrt {\textit {\_C1} \,a^{2}-a^{2} b -\textit {\_C1} a +a b}\, x +\textit {\_C1} a +a^{2} b -a b}}{a}, y \left (x \right ) = -\frac {\sqrt {-a^{2} x^{2}+2 a \sqrt {\textit {\_C1} \,a^{2}-a^{2} b -\textit {\_C1} a +a b}\, x +\textit {\_C1} a +a^{2} b -a b}}{a}\right ]\] 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))
Maple raw output
[y(x) = (-a*x^2+b)^(1/2), y(x) = -(-a*x^2+b)^(1/2), y(x) = 1/a*(-a^2*x^2-2*a*(_C1*a^2-a^2*b-_C1*a+a*b)^(1/2)*x+_C1*a+a^2*b-a*b)^(1/2), y(x) = 1/a*(-a^2*x^2+2*a*(_C1*a^2-a^2*b-_C1*a+a*b)^(1/2)*x+_C1*a+a^2*b-a*b)^(1/2), y(x) = -1/a*(-a^2*x^2-2*a*(_C1*a^2-a^2*b-_C1*a+a*b)^(1/2)*x+_C1*a+a^2*b-a*b)^(1/2), y(x) = -1/a*(-a^2*x^2+2*a*(_C1*a^2-a^2*b-_C1*a+a*b)^(1/2)*x+_C1*a+a^2*b-a*b)^(1/2)]