ODE
\[ y'(x)^n=f(x) (y(x)-a)^{n-1} (y(x)-b)^{n-1} \] ODE Classification
[_separable]
Book solution method
Binomial equation \((y')^m + F(x) G(y)=0\)
Mathematica ✓
cpu = 0.453712 (sec), leaf count = 111
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [-n (a-\text {$\#$1})^{\frac {1}{n}} (\text {$\#$1}-b)^{\frac {1}{n}-1} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{\frac {n-1}{n}} \, _2F_1\left (\frac {1}{n},\frac {n-1}{n};1+\frac {1}{n};\frac {a-\text {$\#$1}}{a-b}\right )\& \right ]\left [\int _1^x(-1)^{\frac {n-1}{n}} f(K[1])^{\frac {1}{n}}dK[1]+c_1\right ]\right \}\right \}\]
Maple ✓
cpu = 2.079 (sec), leaf count = 50
\[\left [\int f \left (x \right )^{\frac {1}{n}}d x -\left (\int _{}^{y \left (x \right )}\frac {\left (\left (-\textit {\_a} +a \right ) \left (-\textit {\_a} +b \right )\right )^{\frac {1}{n}}}{\left (-\textit {\_a} +a \right ) \left (-\textit {\_a} +b \right )}d \textit {\_a} \right )+\textit {\_C1} = 0\right ]\] Mathematica raw input
DSolve[y'[x]^n == f[x]*(-a + y[x])^(-1 + n)*(-b + y[x])^(-1 + n),y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[-(n*Hypergeometric2F1[n^(-1), (-1 + n)/n, 1 + n^(-1), (a - #1)/(a - b)]*(a - #1)^n^(-1)*(-b + #1)^(-1 + n^(-1))*((-b + #1)/(a - b))^((-1 + n)/n)) & ][C[1] + Inactive[Integrate][(-1)^((-1 + n)/n)*f[K[1]]^n^(-1), {K[1], 1, x}]]}}
Maple raw input
dsolve(diff(y(x),x)^n = f(x)*(y(x)-a)^(n-1)*(y(x)-b)^(n-1), y(x))
Maple raw output
[Int(f(x)^(1/n),x)-Intat(1/(-_a+a)/(-_a+b)*((-_a+a)*(-_a+b))^(1/n),_a = y(x))+_C1 = 0]