4.4.21 \(x y'(x)=a x^{2 n}+y(x) (b y(x)+n)\)

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

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