4.4.20 \(x y'(x)=a x^2+b y(x)^2+y(x)\)

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

[[_homogeneous, `class D`], _rational, _Riccati]

Book solution method
Riccati ODE, Special cases

Mathematica
cpu = 0.0197641 (sec), leaf count = 33

\[\left \{\left \{y(x)\to \frac {\sqrt {a} x \tan \left (\sqrt {a} \sqrt {b} \left (c_1+x\right )\right )}{\sqrt {b}}\right \}\right \}\]

Maple
cpu = 0.009 (sec), leaf count = 29

\[ \left \{ {1\arctan \left ({\frac {by \relax (x ) }{x}{\frac {1}{\sqrt {ab}}}} \right ) {\frac {1}{\sqrt {ab}}}}-x-{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[x*y'[x] == a*x^2 + y[x] + b*y[x]^2,y[x],x]

Mathematica raw output

{{y[x] -> (Sqrt[a]*x*Tan[Sqrt[a]*Sqrt[b]*(x + C[1])])/Sqrt[b]}}

Maple raw input

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

Maple raw output

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