4.2.45 \(y'(x)=a x^{\frac {n}{1-n}}+b y(x)^n\)

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

[[_homogeneous, `class G`], _Chini]

Book solution method
Change of Variable, new dependent variable

Mathematica
cpu = 172.366 (sec), leaf count = 109

\[\text {Solve}\left [\int _1^x a K[2]^{\frac {n}{1-n}} \left (\frac {b K[2]^{\frac {n}{n-1}}}{a}\right )^{\frac {1}{n}} \, dK[2]+c_1=\int _1^{y(x) \left (\frac {b x^{\frac {n}{n-1}}}{a}\right )^{\frac {1}{n}}} \frac {1}{-K[1] \left (\frac {(-1)^n (n-1)^{-n} a^{1-n}}{b}\right )^{\frac {1}{n}}+K[1]^n+1} \, dK[1],y(x)\right ]\]

Maple
cpu = 0.305 (sec), leaf count = 61

\[ \left \{ -\int _{{\it \_b}}^{y \relax (x ) }\!{1{x}^{{\frac {n}{n-1}}} \left (\left (bx \left (n-1 \right ) {{\it \_a}}^{n}+{\it \_a} \right ) {x}^{{\frac {n}{n-1}}}+a \left (n-1 \right ) x \right ) ^{-1}}\,{\rm d}{\it \_a} \left (n-1 \right ) +\ln \relax (x ) -{\it \_C1}=0 \right \} \] Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

-Int(1/((b*x*(n-1)*_a^n+_a)*x^(1/(n-1)*n)+a*(n-1)*x)*x^(1/(n-1)*n),_a = _b .. y(
x))*(n-1)+ln(x)-_C1 = 0