4.8.39 \(x^n y'(x)=x^{n-1} \left (a x^{2 n}-b y(x)^2+n y(x)\right )\)

ODE
\[ x^n y'(x)=x^{n-1} \left (a x^{2 n}-b y(x)^2+n y(x)\right ) \] ODE Classification

[_rational, _Riccati]

Book solution method
Change of Variable, new dependent variable

Mathematica
cpu = 0.0217505 (sec), leaf count = 113

\[\left \{\left \{y(x)\to \frac {\sqrt {a} x^n \left (c_1 \sin \left (\frac {\sqrt {a} \sqrt {-b} x^n}{n}\right )-\cos \left (\frac {\sqrt {a} \sqrt {-b} x^n}{n}\right )\right )}{\sqrt {-b} \left (c_1 \cos \left (\frac {\sqrt {a} \sqrt {-b} x^n}{n}\right )+\sin \left (\frac {\sqrt {a} \sqrt {-b} x^n}{n}\right )\right )}\right \}\right \}\]

Maple
cpu = 0.013 (sec), leaf count = 38

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

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

Mathematica raw output

{{y[x] -> (Sqrt[a]*x^n*(-Cos[(Sqrt[a]*Sqrt[-b]*x^n)/n] + C[1]*Sin[(Sqrt[a]*Sqrt[
-b]*x^n)/n]))/(Sqrt[-b]*(C[1]*Cos[(Sqrt[a]*Sqrt[-b]*x^n)/n] + Sin[(Sqrt[a]*Sqrt[
-b]*x^n)/n]))}}

Maple raw input

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

Maple raw output

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