##### 4.12.41 $$x y(x) \left (a+b x^2\right ) y'(x)=A+B y(x)^2$$

ODE
$x y(x) \left (a+b x^2\right ) y'(x)=A+B y(x)^2$ ODE Classiﬁcation

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica
cpu = 0.480925 (sec), leaf count = 98

$\left \{\left \{y(x)\to -\frac {\sqrt {-A+e^{2 B c_1} x^{\frac {2 B}{a}} \left (a+b x^2\right )^{-\frac {B}{a}}}}{\sqrt {B}}\right \},\left \{y(x)\to \frac {\sqrt {-A+e^{2 B c_1} x^{\frac {2 B}{a}} \left (a+b x^2\right )^{-\frac {B}{a}}}}{\sqrt {B}}\right \}\right \}$

Maple
cpu = 0.071 (sec), leaf count = 82

$\left [y \left (x \right ) = \frac {\sqrt {-B \left (-x^{\frac {2 B}{a}} \left (b \,x^{2}+a \right )^{-\frac {B}{a}} \textit {\_C1} B +A \right )}}{B}, y \left (x \right ) = -\frac {\sqrt {-B \left (-x^{\frac {2 B}{a}} \left (b \,x^{2}+a \right )^{-\frac {B}{a}} \textit {\_C1} B +A \right )}}{B}\right ]$ Mathematica raw input

DSolve[x*(a + b*x^2)*y[x]*y'[x] == A + B*y[x]^2,y[x],x]

Mathematica raw output

{{y[x] -> -(Sqrt[-A + (E^(2*B*C[1])*x^((2*B)/a))/(a + b*x^2)^(B/a)]/Sqrt[B])}, {
y[x] -> Sqrt[-A + (E^(2*B*C[1])*x^((2*B)/a))/(a + b*x^2)^(B/a)]/Sqrt[B]}}

Maple raw input

dsolve(x*y(x)*(b*x^2+a)*diff(y(x),x) = A+B*y(x)^2, y(x))

Maple raw output

[y(x) = 1/B*(-B*(-x^(2*B/a)*(b*x^2+a)^(-B/a)*_C1*B+A))^(1/2), y(x) = -1/B*(-B*(-
x^(2*B/a)*(b*x^2+a)^(-B/a)*_C1*B+A))^(1/2)]