4.23.14 \(y'(x)^n=f(x) (y(x)-a)^{n-1} (y(x)-b)^{n-1}\)

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 = 0.532096 (sec), leaf count = 164

\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\frac {n (a-\text {$\#$1})^{\frac {1}{n}} (\text {$\#$1}-b)^{\frac {1}{n}} \left (\frac {a-\text {$\#$1}}{a-b}\right )^{-1/n} \left ((n+1) (a-b) \, _2F_1\left (-\frac {1}{n},\frac {1}{n};1+\frac {1}{n};\frac {\text {$\#$1}-b}{a-b}\right )+(\text {$\#$1}-b) \, _2F_1\left (1+\frac {1}{n},\frac {n-1}{n};2+\frac {1}{n};\frac {\text {$\#$1}-b}{a-b}\right )\right )}{(n+1) (a-b)^2}\& \right ]\left [\int _1^x (-1)^{\frac {n-1}{n}} f(K[1])^{\frac {1}{n}} \, dK[1]+c_1\right ]\right \}\right \}\]

Maple ✓
cpu = 0.338 (sec), leaf count = 50

\[ \left \{ \int \!\sqrt [n]{f \left ( x \right ) }\,{\rm d}x-\int ^{y \left ( x \right ) }\!{\frac {\sqrt [n]{ \left ( -{\it \_a}+a \right ) \left ( -{\it \_a}+b \right ) }}{ \left ( -{\it \_a}+a \right ) \left ( -{\it \_a}+b \right ) }}{d{\it \_a}}+{\it \_C1}=0 \right \} \] 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*(a - #1)^n^(-1)*(-b + #1)^n^(-1)*((a - b)*(1 + n)*H
ypergeometric2F1[-n^(-1), n^(-1), 1 + n^(-1), (-b + #1)/(a - b)] + Hypergeometri
c2F1[1 + n^(-1), (-1 + n)/n, 2 + n^(-1), (-b + #1)/(a - b)]*(-b + #1)))/((a - b)
^2*(1 + n)*((a - #1)/(a - b))^n^(-1)) & ][C[1] + 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),'implicit')

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