##### 4.2.31 $$y'(x)=y(x) \left (a+b y(x)^2\right )$$

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

[_quadrature]

Book solution method
The Bernoulli ODE

Mathematica
cpu = 0.245403 (sec), leaf count = 77

$\left \{\left \{y(x)\to -\frac {i \sqrt {a} e^{a (x+c_1)}}{\sqrt {-1+b e^{2 a (x+c_1)}}}\right \},\left \{y(x)\to \frac {i \sqrt {a} e^{a (x+c_1)}}{\sqrt {-1+b e^{2 a (x+c_1)}}}\right \}\right \}$

Maple
cpu = 0.025 (sec), leaf count = 70

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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