4.12.41 $xy(x)(a+b{x}^2)y^{\prime }(x)=A+By(x{)}^2$

ODE
$$xy(x)(a+b{x}^2)y^{\prime }(x)=A+By(x{)}^2$$ ODE Classification

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica ✓
cpu = 0.0349639 (sec), leaf count = 98

$$\{\{y(x)\to -\frac{\sqrt{e^{2B{c}_1}{x}^{\frac{2B}{a}}{(a+b{x}^2)}^{-\frac{B}{a}}-A}}{\sqrt{B}}\},\{y(x)\to \frac{\sqrt{e^{2B{c}_1}{x}^{\frac{2B}{a}}{(a+b{x}^2)}^{-\frac{B}{a}}-A}}{\sqrt{B}}\}\}$$

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

$$\{\frac{A}{B}-x^{2\frac{B}{a}}{(b{x}^2+a)}^{-\frac{B}{a}}\mathit{\_}\mathit{C}\mathit{1}+{\left(y\left(x\right)\right)}^2=0\}$$ 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