ODE
\[ x y'(x)=a x^{2 n}+y(x) (b y(x)+n) \] ODE Classification
[_rational, _Riccati]
Book solution method
Riccati ODE, Special cases
Mathematica ✓
cpu = 0.0190239 (sec), leaf count = 103
\[\left \{\left \{y(x)\to \frac {\sqrt {a} x^n \left (c_1 \sin \left (\frac {\sqrt {a} \sqrt {b} x^n}{n}\right )-\cos \left (\frac {\sqrt {a} \sqrt {b} x^n}{n}\right )\right )}{\sqrt {b} \left (c_1 \cos \left (\frac {\sqrt {a} \sqrt {b} x^n}{n}\right )+\sin \left (\frac {\sqrt {a} \sqrt {b} x^n}{n}\right )\right )}\right \}\right \}\]
Maple ✓
cpu = 0.014 (sec), leaf count = 34
\[ \left \{ \arctan \left ( {{x}^{-n}y \left ( x \right ) \sqrt {b}{\frac {1}{\sqrt {a}}}} \right ) -{\frac {{x}^{n}}{n}\sqrt {b}\sqrt {a}}+{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[x*y'[x] == a*x^(2*n) + y[x]*(n + b*y[x]),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*diff(y(x),x) = a*x^(2*n)+(n+b*y(x))*y(x), y(x),'implicit')
Maple raw output
arctan(b^(1/2)/a^(1/2)*x^(-n)*y(x))-x^n/n*b^(1/2)*a^(1/2)+_C1 = 0