4.193   xy ′(x) = ay(x)+ b(x2 + 1) y(x)3

ODE

xy′(x) = ay(x) +b (x2 + 1)y(x)3

ODE Classification

[_rational, _Bernoulli]

Book solution method
The Bernoulli ODE

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

{{                                     }  {                                   } }
                    i√a-√a-+-1xa                           i√a-√a-+-1xa
   y(x) → −∘---2a---2------------------  , y(x) → ∘---2a---2------------------
             bx  (ax + a +1) − a(a + 1)c1            bx  (ax + a+ 1)− a(a +1)c1

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

{                   b(ax2 + a+ 1)   }
 − -C12-+ (y (x))−2 + -------------= 0
   (xa)               a (1 + a)

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