4.4.43 \(x y'(x)=a y(x)+b \left (x^2+1\right ) y(x)^3\)

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

[_rational, _Bernoulli]

Book solution method
The Bernoulli ODE

Mathematica
cpu = 0.0348685 (sec), leaf count = 103

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

Maple
cpu = 0.011 (sec), leaf count = 34

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

-1/(x^a)^2*_C1+1/y(x)^2+b*(a*x^2+a+1)/a/(1+a) = 0