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 = 0.0384604 (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 = 0.012 (sec), leaf count = 37

\[ \left \{ {1\arctan \left ({bx{\frac {1}{\sqrt {ab}}}} \right ) {\frac {1}{\sqrt {ab}}}}-{1\arctan \left ({By \relax (x ) {\frac {1}{\sqrt {AB}}}} \right ) {\frac {1}{\sqrt {AB}}}}+{\it \_C1}=0 \right \} \] 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),'implicit')

Maple raw output

1/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))-1/(A*B)^(1/2)*arctan(y(x)*B/(A*B)^(1/2))+_
C1 = 0