##### 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 = 0.446781 (sec), leaf count = 111

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

Maple
cpu = 0.426 (sec), leaf count = 61

$\left [-\left (\int _{\textit {\_b}}^{y \left (x \right )}\frac {x^{\frac {n}{n -1}}}{\left (b x \left (n -1\right ) \textit {\_a}^{n}+\textit {\_a} \right ) x^{\frac {n}{n -1}}+a x \left (n -1\right )}d \textit {\_a} \right ) \left (n -1\right )+\ln \left (x \right )-\textit {\_C1} = 0\right ]$ Mathematica raw input

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

Mathematica raw output

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

Maple raw output

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