##### 4.5.41 $$2 x y'(x)=y(x) \left (y(x)^2+1\right )$$

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

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica
cpu = 0.258518 (sec), leaf count = 63

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

Maple
cpu = 0.016 (sec), leaf count = 42

$\left [y \left (x \right ) = \frac {\sqrt {\left (-x +\textit {\_C1} \right ) x}}{-x +\textit {\_C1}}, y \left (x \right ) = -\frac {\sqrt {\left (-x +\textit {\_C1} \right ) x}}{-x +\textit {\_C1}}\right ]$ Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

[y(x) = 1/(-x+_C1)*((-x+_C1)*x)^(1/2), y(x) = -1/(-x+_C1)*((-x+_C1)*x)^(1/2)]