4.4.40 \(x y'(x)=y(x) \left (y(x)^2+1\right )\)

ODE
\[ x y'(x)=y(x) \left (y(x)^2+1\right ) \] ODE Classification

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica
cpu = 0.0197654 (sec), leaf count = 59

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

Maple
cpu = 0.005 (sec), leaf count = 15

\[ \left \{ 1-{\frac {{\it \_C1}}{{x}^{2}}}+ \left (y \relax (x ) \right ) ^{-2}=0 \right \} \] Mathematica raw input

DSolve[x*y'[x] == y[x]*(1 + y[x]^2),y[x],x]

Mathematica raw output

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

Maple raw input

dsolve(x*diff(y(x),x) = y(x)*(1+y(x)^2), y(x),'implicit')

Maple raw output

1-1/x^2*_C1+1/y(x)^2 = 0