ODE
\[ f(x) (y(x)-a)^2 (y(x)-b)+y'(x)^2=0 \] ODE Classification
[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
Book solution method
Binomial equation \((y')^m + F(x) G(y)=0\)
Mathematica ✓
cpu = 0.288652 (sec), leaf count = 93
\[\left \{\left \{y(x)\to b+(b-a) \tan ^2\left (\frac {1}{2} \sqrt {a-b} \left (\int _1^x-\sqrt {f(K[1])}dK[1]+c_1\right )\right )\right \},\left \{y(x)\to b+(b-a) \tan ^2\left (\frac {1}{2} \sqrt {a-b} \left (\int _1^x\sqrt {f(K[2])}dK[2]+c_1\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.237 (sec), leaf count = 110
\[\left [\frac {2 \arctan \left (\frac {\sqrt {y \left (x \right )-b}}{\sqrt {b -a}}\right )}{\sqrt {b -a}}+\int _{}^{x}\frac {\sqrt {f \left (\textit {\_a} \right ) \left (b -y \left (x \right )\right )}}{\sqrt {y \left (x \right )-b}}d \textit {\_a} +\textit {\_C1} = 0, \frac {2 \arctan \left (\frac {\sqrt {y \left (x \right )-b}}{\sqrt {b -a}}\right )}{\sqrt {b -a}}+\int _{}^{x}-\frac {\sqrt {f \left (\textit {\_a} \right ) \left (b -y \left (x \right )\right )}}{\sqrt {y \left (x \right )-b}}d \textit {\_a} +\textit {\_C1} = 0\right ]\] Mathematica raw input
DSolve[f[x]*(-a + y[x])^2*(-b + y[x]) + y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> b + (-a + b)*Tan[(Sqrt[a - b]*(C[1] + Inactive[Integrate][-Sqrt[f[K[1]]], {K[1], 1, x}]))/2]^2}, {y[x] -> b + (-a + b)*Tan[(Sqrt[a - b]*(C[1] + Inactive[Integrate][Sqrt[f[K[2]]], {K[2], 1, x}]))/2]^2}}
Maple raw input
dsolve(diff(y(x),x)^2+f(x)*(y(x)-a)^2*(y(x)-b) = 0, y(x))
Maple raw output
[2/(b-a)^(1/2)*arctan((y(x)-b)^(1/2)/(b-a)^(1/2))+Intat((f(_a)*(b-y(x)))^(1/2)/(y(x)-b)^(1/2),_a = x)+_C1 = 0, 2/(b-a)^(1/2)*arctan((y(x)-b)^(1/2)/(b-a)^(1/2))+Intat(-(f(_a)*(b-y(x)))^(1/2)/(y(x)-b)^(1/2),_a = x)+_C1 = 0]