4.4.34 \(\frac {1}{2} (n-m) y(x)+x^m+x^n y(x)^2+x y'(x)=0\)

ODE
\[ \frac {1}{2} (n-m) y(x)+x^m+x^n y(x)^2+x y'(x)=0 \] ODE Classification

[_rational, _Riccati]

Book solution method
Riccati ODE, Generalized ODE

Mathematica
cpu = 0.0366563 (sec), leaf count = 40

\[\left \{\left \{y(x)\to -x^{\frac {m-n}{2}} \tan \left (\frac {2 x^{\frac {m+n}{2}}}{m+n}-c_1\right )\right \}\right \}\]

Maple
cpu = 0.015 (sec), leaf count = 34

\[ \left \{ \arctan \left ({x}^{{\frac {n}{2}}-{\frac {m}{2}}}y \relax (x ) \right ) +2\,{\frac {{x}^{n/2+m/2}}{n+m}}+{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[x^m + ((-m + n)*y[x])/2 + x^n*y[x]^2 + x*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> -(x^((m - n)/2)*Tan[(2*x^((m + n)/2))/(m + n) - C[1]])}}

Maple raw input

dsolve(x*diff(y(x),x)+x^m+1/2*(n-m)*y(x)+x^n*y(x)^2 = 0, y(x),'implicit')

Maple raw output

arctan(x^(1/2*n-1/2*m)*y(x))+2*x^(1/2*n+1/2*m)/(n+m)+_C1 = 0