4.16.23 \(f(x) (y(x)-\text {a1}) (y(x)-\text {a2}) (y(x)-\text {a3}) (y(x)-\text {a4})+y'(x)^2=0\)

ODE
\[ f(x) (y(x)-\text {a1}) (y(x)-\text {a2}) (y(x)-\text {a3}) (y(x)-\text {a4})+y'(x)^2=0 \] 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 = 2.85498 (sec), leaf count = 413

\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\frac {2 \sqrt {\text {a2}-\text {$\#$1}} \sqrt {\text {a4}-\text {$\#$1}} \sqrt {\frac {(\text {$\#$1}-\text {a3}) (\text {a1}-\text {a2})}{(\text {$\#$1}-\text {a1}) (\text {a3}-\text {a2})}} F\left (\sin ^{-1}\left (\sqrt {\frac {(\text {a1}-\text {a4}) (\text {$\#$1}-\text {a2})}{(\text {a2}-\text {a4}) (\text {$\#$1}-\text {a1})}}\right )|\frac {(\text {a1}-\text {a3}) (\text {a2}-\text {a4})}{(\text {a2}-\text {a3}) (\text {a1}-\text {a4})}\right )}{\sqrt {\text {a1}-\text {$\#$1}} \sqrt {\text {a3}-\text {$\#$1}} (\text {a2}-\text {a4}) \sqrt {\frac {(\text {a2}-\text {$\#$1}) (\text {$\#$1}-\text {a4}) (\text {a1}-\text {a2}) (\text {a1}-\text {a4})}{(\text {a1}-\text {$\#$1})^2 (\text {a2}-\text {a4})^2}}}\& \right ]\left [c_1+\int _1^x -i \sqrt {f(K[1])} \, dK[1]\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {2 \sqrt {\text {a2}-\text {$\#$1}} \sqrt {\text {a4}-\text {$\#$1}} \sqrt {\frac {(\text {$\#$1}-\text {a3}) (\text {a1}-\text {a2})}{(\text {$\#$1}-\text {a1}) (\text {a3}-\text {a2})}} F\left (\sin ^{-1}\left (\sqrt {\frac {(\text {a1}-\text {a4}) (\text {$\#$1}-\text {a2})}{(\text {a2}-\text {a4}) (\text {$\#$1}-\text {a1})}}\right )|\frac {(\text {a1}-\text {a3}) (\text {a2}-\text {a4})}{(\text {a2}-\text {a3}) (\text {a1}-\text {a4})}\right )}{\sqrt {\text {a1}-\text {$\#$1}} \sqrt {\text {a3}-\text {$\#$1}} (\text {a2}-\text {a4}) \sqrt {\frac {(\text {a2}-\text {$\#$1}) (\text {$\#$1}-\text {a4}) (\text {a1}-\text {a2}) (\text {a1}-\text {a4})}{(\text {a1}-\text {$\#$1})^2 (\text {a2}-\text {a4})^2}}}\& \right ]\left [c_1+\int _1^x i \sqrt {f(K[2])} \, dK[2]\right ]\right \}\right \}\]

Maple
cpu = 0.242 (sec), leaf count = 190

\[ \left \{ \int ^{y \relax (x ) }\!{\frac {1}{\sqrt { \left (-{\it \_a}+{\it a4} \right ) \left (-{\it \_a}+{\it a3} \right ) \left (-{\it \_a}+{\it a2} \right ) \left (-{\it \_a}+{\it a1} \right ) }}}{d{\it \_a}}+\int ^{x}\!{1\sqrt {-f \left ({\it \_a} \right ) \left ({\it a4}-y \relax (x ) \right ) \left ({\it a3}-y \relax (x ) \right ) \left ({\it a2}-y \relax (x ) \right ) \left ({\it a1}-y \relax (x ) \right ) }{\frac {1}{\sqrt { \left ({\it a4}-y \relax (x ) \right ) \left ({\it a3}-y \relax (x ) \right ) \left ({\it a2}-y \relax (x ) \right ) \left ({\it a1}-y \relax (x ) \right ) }}}}{d{\it \_a}}+{\it \_C1}=0,\int ^{y \relax (x ) }\!{\frac {1}{\sqrt { \left (-{\it \_a}+{\it a4} \right ) \left (-{\it \_a}+{\it a3} \right ) \left (-{\it \_a}+{\it a2} \right ) \left (-{\it \_a}+{\it a1} \right ) }}}{d{\it \_a}}+\int ^{x}\!-{1\sqrt {-f \left ({\it \_a} \right ) \left ({\it a4}-y \relax (x ) \right ) \left ({\it a3}-y \relax (x ) \right ) \left ({\it a2}-y \relax (x ) \right ) \left ({\it a1}-y \relax (x ) \right ) }{\frac {1}{\sqrt { \left ({\it a4}-y \relax (x ) \right ) \left ({\it a3}-y \relax (x ) \right ) \left ({\it a2}-y \relax (x ) \right ) \left ({\it a1}-y \relax (x ) \right ) }}}}{d{\it \_a}}+{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[f[x]*(-a1 + y[x])*(-a2 + y[x])*(-a3 + y[x])*(-a4 + y[x]) + y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> InverseFunction[(2*EllipticF[ArcSin[Sqrt[((a1 - a4)*(-a2 + #1))/((a2 -
 a4)*(-a1 + #1))]], ((a1 - a3)*(a2 - a4))/((a2 - a3)*(a1 - a4))]*Sqrt[a2 - #1]*S
qrt[a4 - #1]*Sqrt[((a1 - a2)*(-a3 + #1))/((-a2 + a3)*(-a1 + #1))])/((a2 - a4)*Sq
rt[a1 - #1]*Sqrt[a3 - #1]*Sqrt[((a1 - a2)*(a1 - a4)*(a2 - #1)*(-a4 + #1))/((a2 -
 a4)^2*(a1 - #1)^2)]) & ][C[1] + Integrate[(-I)*Sqrt[f[K[1]]], {K[1], 1, x}]]}, 
{y[x] -> InverseFunction[(2*EllipticF[ArcSin[Sqrt[((a1 - a4)*(-a2 + #1))/((a2 - 
a4)*(-a1 + #1))]], ((a1 - a3)*(a2 - a4))/((a2 - a3)*(a1 - a4))]*Sqrt[a2 - #1]*Sq
rt[a4 - #1]*Sqrt[((a1 - a2)*(-a3 + #1))/((-a2 + a3)*(-a1 + #1))])/((a2 - a4)*Sqr
t[a1 - #1]*Sqrt[a3 - #1]*Sqrt[((a1 - a2)*(a1 - a4)*(a2 - #1)*(-a4 + #1))/((a2 - 
a4)^2*(a1 - #1)^2)]) & ][C[1] + Integrate[I*Sqrt[f[K[2]]], {K[2], 1, x}]]}}

Maple raw input

dsolve(diff(y(x),x)^2+f(x)*(y(x)-a1)*(y(x)-a2)*(y(x)-a3)*(y(x)-a4) = 0, y(x),'implicit')

Maple raw output

Intat(1/((-_a+a4)*(-_a+a3)*(-_a+a2)*(-_a+a1))^(1/2),_a = y(x))+Intat(-(-f(_a)*(a
4-y(x))*(a3-y(x))*(a2-y(x))*(a1-y(x)))^(1/2)/((a4-y(x))*(a3-y(x))*(a2-y(x))*(a1-
y(x)))^(1/2),_a = x)+_C1 = 0, Intat(1/((-_a+a4)*(-_a+a3)*(-_a+a2)*(-_a+a1))^(1/2
),_a = y(x))+Intat((-f(_a)*(a4-y(x))*(a3-y(x))*(a2-y(x))*(a1-y(x)))^(1/2)/((a4-y
(x))*(a3-y(x))*(a2-y(x))*(a1-y(x)))^(1/2),_a = x)+_C1 = 0