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