ODE
\[ y'(x)^n=f(x) (y(x)-a)^{n+1} \] ODE Classification
[_separable]
Book solution method
Binomial equation \((y')^m + F(x) G(y)=0\)
Mathematica ✓
cpu = 0.127541 (sec), leaf count = 41
\[\left \{\left \{y(x)\to a-\left (\frac {\int _1^x (-1)^{\frac {1}{n}+1} f(K[1])^{\frac {1}{n}} \, dK[1]+c_1}{n}\right ){}^{-n}\right \}\right \}\]
Maple ✓
cpu = 0.058 (sec), leaf count = 38
\[ \left \{ -n \left ( a-y \left ( x \right ) \right ) \left ( y \left ( x \right ) -a \right ) ^{{\frac {-n-1}{n}}}+{\it \_C1}+\int \!\sqrt [n]{f \left ( x \right ) }\,{\rm d}x=0 \right \} \] Mathematica raw input
DSolve[y'[x]^n == f[x]*(-a + y[x])^(1 + n),y[x],x]
Mathematica raw output
{{y[x] -> a - ((C[1] + Integrate[(-1)^(1 + n^(-1))*f[K[1]]^n^(-1), {K[1], 1, x}])/n)^(-n)}}
Maple raw input
dsolve(diff(y(x),x)^n = f(x)*(y(x)-a)^(n+1), y(x),'implicit')
Maple raw output
-n*(a-y(x))*(y(x)-a)^((-n-1)/n)+_C1+Int(f(x)^(1/n),x) = 0