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

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

[_quadrature]

Book solution method
Missing Variables ODE, Independent variable missing, Solve for \(y'\)

Mathematica
cpu = 0.602337 (sec), leaf count = 173

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

Maple
cpu = 0.159 (sec), leaf count = 90

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

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

Mathematica raw output

{{y[x] -> JacobiNS[(Sqrt[a - b]*((-I)*x + C[1]))/2, (a - c)/(a - b)]^2*(-a + b +
 a*JacobiSN[(Sqrt[a - b]*((-I)*x + C[1]))/2, (a - c)/(a - b)]^2)}, {y[x] -> Jaco
biNS[(Sqrt[a - b]*(I*x + C[1]))/2, (a - c)/(a - b)]^2*(-a + b + a*JacobiSN[(Sqrt
[a - b]*(I*x + C[1]))/2, (a - c)/(a - b)]^2)}}

Maple raw input

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

Maple raw output

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