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 = 0.342113 (sec), leaf count = 81
\[\left \{\left \{y(x)\to \frac {b n^n+a (a-b)^n \left (\int _1^x(-1)^{1+\frac {1}{n}} f(K[1])dK[1]+c_1\right ){}^n}{n^n+(a-b)^n \left (\int _1^x(-1)^{1+\frac {1}{n}} f(K[1])dK[1]+c_1\right ){}^n}\right \}\right \}\]
Maple ✓
cpu = 4.726 (sec), leaf count = 127
\[\left [y \left (x \right ) = \frac {\left (\frac {n}{-\textit {\_C1} a +\textit {\_C1} b -a \left (\int f \left (x \right )d x \right )+b \left (\int f \left (x \right )d x \right )}\right )^{n} b}{-1+\left (\frac {n}{-\textit {\_C1} a +\textit {\_C1} b -a \left (\int f \left (x \right )d x \right )+b \left (\int f \left (x \right )d x \right )}\right )^{n}}-\frac {\left (\frac {n}{-\textit {\_C1} a +\textit {\_C1} b -a \left (\int f \left (x \right )d x \right )+b \left (\int f \left (x \right )d x \right )}\right )^{n} a}{-1+\left (\frac {n}{-\textit {\_C1} a +\textit {\_C1} b -a \left (\int f \left (x \right )d x \right )+b \left (\int f \left (x \right )d x \right )}\right )^{n}}+a\right ]\] 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] + Inactive[Integrate][(-1)^(1 + n^(-1))*f[K[1]], {K[1], 1, x}])^n)/(n^n + (a - b)^n*(C[1] + Inactive[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))
Maple raw output
[y(x) = (n/(-_C1*a+_C1*b-a*Int(f(x),x)+b*Int(f(x),x)))^n/(-1+(n/(-_C1*a+_C1*b-a*Int(f(x),x)+b*Int(f(x),x)))^n)*b-(n/(-_C1*a+_C1*b-a*Int(f(x),x)+b*Int(f(x),x)))^n/(-1+(n/(-_C1*a+_C1*b-a*Int(f(x),x)+b*Int(f(x),x)))^n)*a+a]