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 = 0.0349639 (sec), leaf count = 98

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

Maple
cpu = 0.02 (sec), leaf count = 38

\[ \left \{ {\frac {A}{B}}-{x}^{2\,{\frac {B}{a}}} \left (b{x}^{2}+a \right ) ^{-{\frac {B}{a}}}{\it \_C1}+ \left (y \relax (x ) \right ) ^{2}=0 \right \} \] 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),'implicit')

Maple raw output

A/B-x^(2*B/a)*(b*x^2+a)^(-B/a)*_C1+y(x)^2 = 0