4.15.23 \(x y'(x) \left (a+x y(x)^n\right )+b y(x)=0\)

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

[[_homogeneous, `class G`], _rational]

Book solution method
Change of Variable, new independent variable

Mathematica
cpu = 0.164489 (sec), leaf count = 50

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

Maple
cpu = 0.085 (sec), leaf count = 37

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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