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

ODE
\[ f(x) (y(x)-a)^2 (y(x)-b) (y(x)-c)+y'(x)^2=0 \] 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 = 0.720837 (sec), leaf count = 931

\[\left \{\left \{y(x)\to \frac {a \left (e^{2 \sqrt {b-a} \sqrt {c-a} \left (c_1+\int _1^x -i \sqrt {f(K[1])} \, dK[1]\right )} b^2-2 e^{\sqrt {b-a} \sqrt {c-a} \left (c_1+\int _1^x -i \sqrt {f(K[1])} \, dK[1]\right )} \left (c e^{\sqrt {b-a} \sqrt {c-a} \left (c_1+\int _1^x -i \sqrt {f(K[1])} \, dK[1]\right )}-1\right ) b+\left (e^{\sqrt {b-a} \sqrt {c-a} \left (c_1+\int _1^x -i \sqrt {f(K[1])} \, dK[1]\right )} c+1\right ){}^2\right )-4 b c e^{\sqrt {b-a} \sqrt {c-a} \left (c_1+\int _1^x -i \sqrt {f(K[1])} \, dK[1]\right )}}{e^{2 \sqrt {b-a} \sqrt {c-a} \left (c_1+\int _1^x -i \sqrt {f(K[1])} \, dK[1]\right )} b^2-2 e^{\sqrt {b-a} \sqrt {c-a} \left (c_1+\int _1^x -i \sqrt {f(K[1])} \, dK[1]\right )} \left (e^{\sqrt {b-a} \sqrt {c-a} \left (c_1+\int _1^x -i \sqrt {f(K[1])} \, dK[1]\right )} c+1\right ) b+4 a e^{\sqrt {b-a} \sqrt {c-a} \left (c_1+\int _1^x -i \sqrt {f(K[1])} \, dK[1]\right )}+\left (c e^{\sqrt {b-a} \sqrt {c-a} \left (c_1+\int _1^x -i \sqrt {f(K[1])} \, dK[1]\right )}-1\right ){}^2}\right \},\left \{y(x)\to \frac {a \left (e^{2 \sqrt {b-a} \sqrt {c-a} \left (c_1+\int _1^x i \sqrt {f(K[2])} \, dK[2]\right )} b^2-2 e^{\sqrt {b-a} \sqrt {c-a} \left (c_1+\int _1^x i \sqrt {f(K[2])} \, dK[2]\right )} \left (c e^{\sqrt {b-a} \sqrt {c-a} \left (c_1+\int _1^x i \sqrt {f(K[2])} \, dK[2]\right )}-1\right ) b+\left (e^{\sqrt {b-a} \sqrt {c-a} \left (c_1+\int _1^x i \sqrt {f(K[2])} \, dK[2]\right )} c+1\right ){}^2\right )-4 b c e^{\sqrt {b-a} \sqrt {c-a} \left (c_1+\int _1^x i \sqrt {f(K[2])} \, dK[2]\right )}}{e^{2 \sqrt {b-a} \sqrt {c-a} \left (c_1+\int _1^x i \sqrt {f(K[2])} \, dK[2]\right )} b^2-2 e^{\sqrt {b-a} \sqrt {c-a} \left (c_1+\int _1^x i \sqrt {f(K[2])} \, dK[2]\right )} \left (e^{\sqrt {b-a} \sqrt {c-a} \left (c_1+\int _1^x i \sqrt {f(K[2])} \, dK[2]\right )} c+1\right ) b+4 a e^{\sqrt {b-a} \sqrt {c-a} \left (c_1+\int _1^x i \sqrt {f(K[2])} \, dK[2]\right )}+\left (c e^{\sqrt {b-a} \sqrt {c-a} \left (c_1+\int _1^x i \sqrt {f(K[2])} \, dK[2]\right )}-1\right ){}^2}\right \}\right \}\]

Maple
cpu = 0.247 (sec), leaf count = 382

\[ \left \{ {\frac {1}{ \left (a-b \right ) \left (a-c \right ) } \left (-\ln \left ({\frac {1}{a-y \relax (x ) } \left (-2\,\sqrt { \left (a-b \right ) \left (a-c \right ) }\sqrt { \left (b-y \relax (x ) \right ) \left (c-y \relax (x ) \right ) }+ \left (-2\,a+b+c \right ) y \relax (x ) + \left (b+c \right ) a-2\,cb \right ) } \right ) \sqrt { \left (a-b \right ) \left (a-c \right ) }\sqrt {y \relax (x ) -c}\sqrt {y \relax (x ) -b}+\sqrt { \left (b-y \relax (x ) \right ) \left (c-y \relax (x ) \right ) } \left (a-c \right ) \left (a-b \right ) \left (\int ^{x}\!{1\sqrt {-f \left ({\it \_a} \right ) \left (c-y \relax (x ) \right ) \left (b-y \relax (x ) \right ) }{\frac {1}{\sqrt {y \relax (x ) -b}}}{\frac {1}{\sqrt {y \relax (x ) -c}}}}{d{\it \_a}}+{\it \_C1} \right ) \right ) {\frac {1}{\sqrt { \left (b-y \relax (x ) \right ) \left (c-y \relax (x ) \right ) }}}}=0,{\frac {1}{ \left (a-b \right ) \left (a-c \right ) } \left (-\ln \left ({\frac {1}{a-y \relax (x ) } \left (-2\,\sqrt { \left (a-b \right ) \left (a-c \right ) }\sqrt { \left (b-y \relax (x ) \right ) \left (c-y \relax (x ) \right ) }+ \left (-2\,a+b+c \right ) y \relax (x ) + \left (b+c \right ) a-2\,cb \right ) } \right ) \sqrt { \left (a-b \right ) \left (a-c \right ) }\sqrt {y \relax (x ) -c}\sqrt {y \relax (x ) -b}+\sqrt { \left (b-y \relax (x ) \right ) \left (c-y \relax (x ) \right ) } \left (a-c \right ) \left (a-b \right ) \left (\int ^{x}\!-{1\sqrt {-f \left ({\it \_a} \right ) \left (c-y \relax (x ) \right ) \left (b-y \relax (x ) \right ) }{\frac {1}{\sqrt {y \relax (x ) -b}}}{\frac {1}{\sqrt {y \relax (x ) -c}}}}{d{\it \_a}}+{\it \_C1} \right ) \right ) {\frac {1}{\sqrt { \left (b-y \relax (x ) \right ) \left (c-y \relax (x ) \right ) }}}}=0 \right \} \] Mathematica raw input

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

Mathematica raw output

{{y[x] -> (-4*b*c*E^(Sqrt[-a + b]*Sqrt[-a + c]*(C[1] + Integrate[(-I)*Sqrt[f[K[1
]]], {K[1], 1, x}])) + a*(b^2*E^(2*Sqrt[-a + b]*Sqrt[-a + c]*(C[1] + Integrate[(
-I)*Sqrt[f[K[1]]], {K[1], 1, x}])) - 2*b*E^(Sqrt[-a + b]*Sqrt[-a + c]*(C[1] + In
tegrate[(-I)*Sqrt[f[K[1]]], {K[1], 1, x}]))*(-1 + c*E^(Sqrt[-a + b]*Sqrt[-a + c]
*(C[1] + Integrate[(-I)*Sqrt[f[K[1]]], {K[1], 1, x}]))) + (1 + c*E^(Sqrt[-a + b]
*Sqrt[-a + c]*(C[1] + Integrate[(-I)*Sqrt[f[K[1]]], {K[1], 1, x}])))^2))/(4*a*E^
(Sqrt[-a + b]*Sqrt[-a + c]*(C[1] + Integrate[(-I)*Sqrt[f[K[1]]], {K[1], 1, x}]))
 + b^2*E^(2*Sqrt[-a + b]*Sqrt[-a + c]*(C[1] + Integrate[(-I)*Sqrt[f[K[1]]], {K[1
], 1, x}])) + (-1 + c*E^(Sqrt[-a + b]*Sqrt[-a + c]*(C[1] + Integrate[(-I)*Sqrt[f
[K[1]]], {K[1], 1, x}])))^2 - 2*b*E^(Sqrt[-a + b]*Sqrt[-a + c]*(C[1] + Integrate
[(-I)*Sqrt[f[K[1]]], {K[1], 1, x}]))*(1 + c*E^(Sqrt[-a + b]*Sqrt[-a + c]*(C[1] +
 Integrate[(-I)*Sqrt[f[K[1]]], {K[1], 1, x}]))))}, {y[x] -> (-4*b*c*E^(Sqrt[-a +
 b]*Sqrt[-a + c]*(C[1] + Integrate[I*Sqrt[f[K[2]]], {K[2], 1, x}])) + a*(b^2*E^(
2*Sqrt[-a + b]*Sqrt[-a + c]*(C[1] + Integrate[I*Sqrt[f[K[2]]], {K[2], 1, x}])) -
 2*b*E^(Sqrt[-a + b]*Sqrt[-a + c]*(C[1] + Integrate[I*Sqrt[f[K[2]]], {K[2], 1, x
}]))*(-1 + c*E^(Sqrt[-a + b]*Sqrt[-a + c]*(C[1] + Integrate[I*Sqrt[f[K[2]]], {K[
2], 1, x}]))) + (1 + c*E^(Sqrt[-a + b]*Sqrt[-a + c]*(C[1] + Integrate[I*Sqrt[f[K
[2]]], {K[2], 1, x}])))^2))/(4*a*E^(Sqrt[-a + b]*Sqrt[-a + c]*(C[1] + Integrate[
I*Sqrt[f[K[2]]], {K[2], 1, x}])) + b^2*E^(2*Sqrt[-a + b]*Sqrt[-a + c]*(C[1] + In
tegrate[I*Sqrt[f[K[2]]], {K[2], 1, x}])) + (-1 + c*E^(Sqrt[-a + b]*Sqrt[-a + c]*
(C[1] + Integrate[I*Sqrt[f[K[2]]], {K[2], 1, x}])))^2 - 2*b*E^(Sqrt[-a + b]*Sqrt
[-a + c]*(C[1] + Integrate[I*Sqrt[f[K[2]]], {K[2], 1, x}]))*(1 + c*E^(Sqrt[-a + 
b]*Sqrt[-a + c]*(C[1] + Integrate[I*Sqrt[f[K[2]]], {K[2], 1, x}]))))}}

Maple raw input

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

Maple raw output

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