4.16.21 \(f(x) (y(x)-a) (y(x)-b) (y(x)-c)+y'(x)^2=0\)

ODE
\[ f(x) (y(x)-a) (y(x)-b) (y(x)-c)+y'(x)^2=0 \] ODE Classification

[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

Book solution method
Binomial equation \((y')^m + F(x) G(y)=0\)

Mathematica
cpu = 0.491013 (sec), leaf count = 209

\[\left \{\left \{y(x)\to \text {ns}\left (\frac {1}{2} \sqrt {a-b} \left (c_1+\int _1^x -\sqrt {f(K[1])} \, dK[1]\right )|\frac {a-c}{a-b}\right ){}^2 \left (a \text {sn}\left (\frac {1}{2} \sqrt {a-b} \left (c_1+\int _1^x -\sqrt {f(K[1])} \, dK[1]\right )|\frac {a-c}{a-b}\right ){}^2-a+b\right )\right \},\left \{y(x)\to \text {ns}\left (\frac {1}{2} \sqrt {a-b} \left (c_1+\int _1^x \sqrt {f(K[2])} \, dK[2]\right )|\frac {a-c}{a-b}\right ){}^2 \left (a \text {sn}\left (\frac {1}{2} \sqrt {a-b} \left (c_1+\int _1^x \sqrt {f(K[2])} \, dK[2]\right )|\frac {a-c}{a-b}\right ){}^2-a+b\right )\right \}\right \}\]

Maple
cpu = 0.195 (sec), leaf count = 158

\[ \left \{ \int ^{y \relax (x ) }\!{\frac {1}{\sqrt {- \left (-{\it \_a}+c \right ) \left (-{\it \_a}+b \right ) \left (-{\it \_a}+a \right ) }}}{d{\it \_a}}+\int ^{x}\!{1\sqrt {f \left ({\it \_a} \right ) \left (c-y \relax (x ) \right ) \left (b-y \relax (x ) \right ) \left (a-y \relax (x ) \right ) }{\frac {1}{\sqrt {- \left (c-y \relax (x ) \right ) \left (b-y \relax (x ) \right ) \left (a-y \relax (x ) \right ) }}}}{d{\it \_a}}+{\it \_C1}=0,\int ^{y \relax (x ) }\!{\frac {1}{\sqrt {- \left (-{\it \_a}+c \right ) \left (-{\it \_a}+b \right ) \left (-{\it \_a}+a \right ) }}}{d{\it \_a}}+\int ^{x}\!-{1\sqrt {f \left ({\it \_a} \right ) \left (c-y \relax (x ) \right ) \left (b-y \relax (x ) \right ) \left (a-y \relax (x ) \right ) }{\frac {1}{\sqrt {- \left (c-y \relax (x ) \right ) \left (b-y \relax (x ) \right ) \left (a-y \relax (x ) \right ) }}}}{d{\it \_a}}+{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[f[x]*(-a + y[x])*(-b + y[x])*(-c + y[x]) + y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> JacobiNS[(Sqrt[a - b]*(C[1] + Integrate[-Sqrt[f[K[1]]], {K[1], 1, x}])
)/2, (a - c)/(a - b)]^2*(-a + b + a*JacobiSN[(Sqrt[a - b]*(C[1] + Integrate[-Sqr
t[f[K[1]]], {K[1], 1, x}]))/2, (a - c)/(a - b)]^2)}, {y[x] -> JacobiNS[(Sqrt[a -
 b]*(C[1] + Integrate[Sqrt[f[K[2]]], {K[2], 1, x}]))/2, (a - c)/(a - b)]^2*(-a +
 b + a*JacobiSN[(Sqrt[a - b]*(C[1] + Integrate[Sqrt[f[K[2]]], {K[2], 1, x}]))/2,
 (a - c)/(a - b)]^2)}}

Maple raw input

dsolve(diff(y(x),x)^2+f(x)*(y(x)-a)*(y(x)-b)*(y(x)-c) = 0, y(x),'implicit')

Maple raw output

Intat(1/(-(-_a+c)*(-_a+b)*(-_a+a))^(1/2),_a = y(x))+Intat(-(f(_a)*(c-y(x))*(b-y(
x))*(a-y(x)))^(1/2)/(-(c-y(x))*(b-y(x))*(a-y(x)))^(1/2),_a = x)+_C1 = 0, Intat(1
/(-(-_a+c)*(-_a+b)*(-_a+a))^(1/2),_a = y(x))+Intat((f(_a)*(c-y(x))*(b-y(x))*(a-y
(x)))^(1/2)/(-(c-y(x))*(b-y(x))*(a-y(x)))^(1/2),_a = x)+_C1 = 0