ODE
\[ f(x) (y(x)-a) (y(x)-b)+y'(x)^2=0 \] 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 = 0.353267 (sec), leaf count = 175
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {2 \sqrt {a-b} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \sin ^{-1}\left (\frac {\sqrt {a-\text {$\#$1}}}{\sqrt {a-b}}\right )}{\sqrt {b-\text {$\#$1}}}\& \right ]\left [\int _1^x-i \sqrt {f(K[2])}dK[2]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {2 \sqrt {a-b} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \sin ^{-1}\left (\frac {\sqrt {a-\text {$\#$1}}}{\sqrt {a-b}}\right )}{\sqrt {b-\text {$\#$1}}}\& \right ]\left [\int _1^xi \sqrt {f(K[3])}dK[3]+c_1\right ]\right \}\right \}\]
Maple ✓
cpu = 0.348 (sec), leaf count = 212
\[\left [\frac {\sqrt {\left (y \left (x \right )-a \right ) \left (y \left (x \right )-b \right )}\, \ln \left (-\frac {a}{2}-\frac {b}{2}+y \left (x \right )+\sqrt {y \left (x \right )^{2}+\left (-a -b \right ) y \left (x \right )+a b}\right )}{\sqrt {y \left (x \right )-b}\, \sqrt {y \left (x \right )-a}}+\int _{}^{x}-\frac {\sqrt {-f \left (\textit {\_a} \right ) \left (a -y \left (x \right )\right ) \left (b -y \left (x \right )\right )}}{\sqrt {y \left (x \right )-b}\, \sqrt {y \left (x \right )-a}}d \textit {\_a} +\textit {\_C1} = 0, \frac {\sqrt {\left (y \left (x \right )-a \right ) \left (y \left (x \right )-b \right )}\, \ln \left (-\frac {a}{2}-\frac {b}{2}+y \left (x \right )+\sqrt {y \left (x \right )^{2}+\left (-a -b \right ) y \left (x \right )+a b}\right )}{\sqrt {y \left (x \right )-b}\, \sqrt {y \left (x \right )-a}}+\int _{}^{x}\frac {\sqrt {-f \left (\textit {\_a} \right ) \left (a -y \left (x \right )\right ) \left (b -y \left (x \right )\right )}}{\sqrt {y \left (x \right )-b}\, \sqrt {y \left (x \right )-a}}d \textit {\_a} +\textit {\_C1} = 0\right ]\] Mathematica raw input
DSolve[f[x]*(-a + y[x])*(-b + y[x]) + y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[(-2*Sqrt[a - b]*ArcSin[Sqrt[a - #1]/Sqrt[a - b]]*Sqrt[(-b + #1)/(a - b)])/Sqrt[b - #1] & ][C[1] + Inactive[Integrate][(-I)*Sqrt[f[K[2]]], {K[2], 1, x}]]}, {y[x] -> InverseFunction[(-2*Sqrt[a - b]*ArcSin[Sqrt[a - #1]/Sqrt[a - b]]*Sqrt[(-b + #1)/(a - b)])/Sqrt[b - #1] & ][C[1] + Inactive[Integrate][I*Sqrt[f[K[3]]], {K[3], 1, x}]]}}
Maple raw input
dsolve(diff(y(x),x)^2+f(x)*(y(x)-a)*(y(x)-b) = 0, y(x))
Maple raw output
[((y(x)-a)*(y(x)-b))^(1/2)/(y(x)-b)^(1/2)/(y(x)-a)^(1/2)*ln(-1/2*a-1/2*b+y(x)+(y(x)^2+(-a-b)*y(x)+a*b)^(1/2))+Intat(-1/(y(x)-b)^(1/2)/(y(x)-a)^(1/2)*(-f(_a)*(a-y(x))*(b-y(x)))^(1/2),_a = x)+_C1 = 0, ((y(x)-a)*(y(x)-b))^(1/2)/(y(x)-b)^(1/2)/(y(x)-a)^(1/2)*ln(-1/2*a-1/2*b+y(x)+(y(x)^2+(-a-b)*y(x)+a*b)^(1/2))+Intat(1/(y(x)-b)^(1/2)/(y(x)-a)^(1/2)*(-f(_a)*(a-y(x))*(b-y(x)))^(1/2),_a = x)+_C1 = 0]