##### 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 = 0.641636 (sec), leaf count = 173

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

Maple
cpu = 0.141 (sec), leaf count = 79

$\left [y \left (x \right ) = a, y \left (x \right ) = b, y \left (x \right ) = c, x -\left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\left (\textit {\_a} -a \right ) \left (\textit {\_a} -b \right ) \left (\textit {\_a} -c \right )}}d \textit {\_a} \right )-\textit {\_C1} = 0, x -\left (\int _{}^{y \left (x \right )}-\frac {1}{\sqrt {\left (\textit {\_a} -a \right ) \left (\textit {\_a} -b \right ) \left (\textit {\_a} -c \right )}}d \textit {\_a} \right )-\textit {\_C1} = 0\right ]$ 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))

Maple raw output

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