4.12.17 \(x y'(x) \left (a y(x)+x^n\right )+y(x)^2 (b+c y(x))=0\)

ODE
\[ x y'(x) \left (a y(x)+x^n\right )+y(x)^2 (b+c y(x))=0 \] ODE Classification

[_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`], [_Abel, `2nd type``class C`]]

Book solution method
Change of Variable, new dependent variable

Mathematica
cpu = 2.35039 (sec), leaf count = 77

\[\text {Solve}\left [-\frac {x^{-n} y(x)^{-\frac {a n+b}{b}} (b+c y(x))^{\frac {a n}{b}} \left (y(x) \left (a^2 n+a b-c x^n\right )+a n x^n\right )}{a^2 n^2 (a n+b)}=c_1,y(x)\right ]\]

Maple
cpu = 0.055 (sec), leaf count = 77

\[ \left \{ {\it \_C1}+{\frac {c-{x}^{-n}ab}{an} \left ({\frac {b+cy \relax (x ) }{by \relax (x ) }} \right ) ^{{\frac {an}{b}}}}-{\frac {b}{an+b} \left ({\frac {b+cy \relax (x ) }{by \relax (x ) }} \right ) ^{{\frac {an}{b}}+1}}=0 \right \} \] Mathematica raw input

DSolve[y[x]^2*(b + c*y[x]) + x*(x^n + a*y[x])*y'[x] == 0,y[x],x]

Mathematica raw output

Solve[-(((b + c*y[x])^((a*n)/b)*(a*n*x^n + (a*b + a^2*n - c*x^n)*y[x]))/(a^2*n^2
*(b + a*n)*x^n*y[x]^((b + a*n)/b))) == C[1], y[x]]

Maple raw input

dsolve(x*(x^n+a*y(x))*diff(y(x),x)+(b+c*y(x))*y(x)^2 = 0, y(x),'implicit')

Maple raw output

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