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 = 0.0339076 (sec), leaf count = 77

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

Maple
cpu = 0.005 (sec), leaf count = 21

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

Maple raw output

1/a*b-exp(-2*a*x)*_C1+1/y(x)^2 = 0