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] -> 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)}}
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]