ODE
\[ y'(x)^n=f(x)^n (y(x)-a)^{n+1} (y(x)-b)^{n-1} \] ODE Classification
[_separable]
Book solution method
Change of variable
Mathematica ✓
cpu = 0.514525 (sec), leaf count = 79
\[\left \{\left \{y(x)\to \frac {a (a-b)^n \left (\int _1^x (-1)^{\frac {1}{n}+1} f(K[1]) \, dK[1]+c_1\right ){}^n+b n^n}{(a-b)^n \left (\int _1^x (-1)^{\frac {1}{n}+1} f(K[1]) \, dK[1]+c_1\right ){}^n+n^n}\right \}\right \}\]
Maple ✓
cpu = 0.269 (sec), leaf count = 70
\[ \left \{ {\frac {1}{a-b} \left ( \left ( y \left ( x \right ) -b \right ) ^{{\frac {1-n}{n}}}n \left ( b-y \left ( x \right ) \right ) \left ( a-y \left ( x \right ) \right ) \left ( y \left ( x \right ) -a \right ) ^{{\frac {-n-1}{n}}}+ \left ( {\it \_C1}+\int \!f \left ( x \right ) \,{\rm d}x \right ) \left ( a-b \right ) \right ) }=0 \right \} \] Mathematica raw input
DSolve[y'[x]^n == f[x]^n*(-a + y[x])^(1 + n)*(-b + y[x])^(-1 + n),y[x],x]
Mathematica raw output
{{y[x] -> (b*n^n + a*(a - b)^n*(C[1] + Integrate[(-1)^(1 + n^(-1))*f[K[1]], {K[1], 1, x}])^n)/(n^n + (a - b)^n*(C[1] + Integrate[(-1)^(1 + n^(-1))*f[K[1]], {K[1], 1, x}])^n)}}
Maple raw input
dsolve(diff(y(x),x)^n = f(x)^n*(y(x)-a)^(n+1)*(y(x)-b)^(n-1), y(x),'implicit')
Maple raw output
((y(x)-b)^((1-n)/n)*n*(b-y(x))*(a-y(x))*(y(x)-a)^((-n-1)/n)+(_C1+Int(f(x),x))*(a-b))/(a-b) = 0