##### 4.12.33 $$2 x (x+1) y(x) y'(x)=y(x)^2+1$$

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

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica
cpu = 0.403901 (sec), leaf count = 58

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

Maple
cpu = 0.026 (sec), leaf count = 42

$\left [y \left (x \right ) = \frac {\sqrt {\left (x +1\right ) \left (x \textit {\_C1} -1\right )}}{x +1}, y \left (x \right ) = -\frac {\sqrt {\left (x +1\right ) \left (x \textit {\_C1} -1\right )}}{x +1}\right ]$ Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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