##### 4.7.42 $$\left (a+b x^2\right ) y'(x)=A+B y(x)^2$$

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

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica
cpu = 0.36262 (sec), leaf count = 55

$\left \{\left \{y(x)\to \frac {\sqrt {A} \tan \left (\sqrt {A} \sqrt {B} \left (\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}+c_1\right )\right )}{\sqrt {B}}\right \}\right \}$

Maple
cpu = 0.061 (sec), leaf count = 42

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

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

Mathematica raw output

{{y[x] -> (Sqrt[A]*Tan[Sqrt[A]*Sqrt[B]*(ArcTan[(Sqrt[b]*x)/Sqrt[a]]/(Sqrt[a]*Sqr
t[b]) + C[1])])/Sqrt[B]}}

Maple raw input

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

Maple raw output

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