4.16.24 \(y'(x)^2=f(x)^2 (y(x)-a) (y(x)-b) (y(x)-c)^2\)

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

[_separable]

Book solution method
Change of variable

Mathematica
cpu = 0.585993 (sec), leaf count = 865

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

Maple
cpu = 0.154 (sec), leaf count = 195

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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