##### 4.4.42 $$x y'(x)+y(x)=a \left (x^2+1\right ) y(x)^3$$

ODE
$x y'(x)+y(x)=a \left (x^2+1\right ) y(x)^3$ ODE Classiﬁcation

[_rational, _Bernoulli]

Book solution method
The Bernoulli ODE

Mathematica
cpu = 0.269855 (sec), leaf count = 51

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

Maple
cpu = 0.024 (sec), leaf count = 43

$\left [y \left (x \right ) = \frac {1}{\sqrt {-2 \ln \left (x \right ) x^{2} a +x^{2} \textit {\_C1} +a}}, y \left (x \right ) = -\frac {1}{\sqrt {-2 \ln \left (x \right ) x^{2} a +x^{2} \textit {\_C1} +a}}\right ]$ Mathematica raw input

DSolve[y[x] + x*y'[x] == a*(1 + x^2)*y[x]^3,y[x],x]

Mathematica raw output

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

Maple raw input

dsolve(x*diff(y(x),x)+y(x) = a*(x^2+1)*y(x)^3, y(x))

Maple raw output

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