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 = 0.0119158 (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 = 0.013 (sec), leaf count = 25

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

Maple raw output

-a+2*x^2*ln(x)*a-x^2*_C1+1/y(x)^2 = 0