4.19.36 \(4 x (a-x) (b-x) y'(x)^2=\left (-2 x (a+b)+a b+2 x^2\right )^2\)

ODE
\[ 4 x (a-x) (b-x) y'(x)^2=\left (-2 x (a+b)+a b+2 x^2\right )^2 \] ODE Classification

[_quadrature]

Book solution method
Change of variable

Mathematica
cpu = 0.955217 (sec), leaf count = 297

\[\left \{\left \{y(x)\to c_1-\frac {i (x-a) \left (2 x (x-b)+\frac {i \sqrt {\frac {x-b}{a-b}} \left (2 \left (a^2-b^2\right ) E\left (i \sinh ^{-1}\left (\sqrt {\frac {x}{a}-1}\right )|\frac {a}{a-b}\right )+b (a+2 b) F\left (i \sinh ^{-1}\left (\sqrt {\frac {x}{a}-1}\right )|\frac {a}{a-b}\right )\right )}{\sqrt {1-\frac {a}{x}}}\right )}{3 \sqrt {-x (x-a) (x-b)}}\right \},\left \{y(x)\to c_1+\frac {i (x-a) \left (2 x (x-b)+\frac {i \sqrt {\frac {x-b}{a-b}} \left (2 \left (a^2-b^2\right ) E\left (i \sinh ^{-1}\left (\sqrt {\frac {x}{a}-1}\right )|\frac {a}{a-b}\right )+b (a+2 b) F\left (i \sinh ^{-1}\left (\sqrt {\frac {x}{a}-1}\right )|\frac {a}{a-b}\right )\right )}{\sqrt {1-\frac {a}{x}}}\right )}{3 \sqrt {-x (x-a) (x-b)}}\right \}\right \}\]

Maple
cpu = 0.069 (sec), leaf count = 83

\[ \left \{ y \relax (x ) =\int \!-{\frac {2\,{x}^{2}+ \left (-2\,a-2\,b \right ) x+ab}{2}{\frac {1}{\sqrt {x \left (b-x \right ) \left (a-x \right ) }}}}\,{\rm d}x+{\it \_C1},y \relax (x ) =\int \!{\frac {2\,{x}^{2}+ \left (-2\,a-2\,b \right ) x+ab}{2}{\frac {1}{\sqrt {x \left (b-x \right ) \left (a-x \right ) }}}}\,{\rm d}x+{\it \_C1} \right \} \] Mathematica raw input

DSolve[4*(a - x)*(b - x)*x*y'[x]^2 == (a*b - 2*(a + b)*x + 2*x^2)^2,y[x],x]

Mathematica raw output

{{y[x] -> C[1] - ((I/3)*(-a + x)*(2*x*(-b + x) + (I*Sqrt[(-b + x)/(a - b)]*(2*(a
^2 - b^2)*EllipticE[I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)] + b*(a + 2*b)*Elliptic
F[I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)]))/Sqrt[1 - a/x]))/Sqrt[-(x*(-a + x)*(-b 
+ x))]}, {y[x] -> C[1] + ((I/3)*(-a + x)*(2*x*(-b + x) + (I*Sqrt[(-b + x)/(a - b
)]*(2*(a^2 - b^2)*EllipticE[I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)] + b*(a + 2*b)*
EllipticF[I*ArcSinh[Sqrt[-1 + x/a]], a/(a - b)]))/Sqrt[1 - a/x]))/Sqrt[-(x*(-a +
 x)*(-b + x))]}}

Maple raw input

dsolve(4*x*(a-x)*(b-x)*diff(y(x),x)^2 = (a*b-2*x*(a+b)+2*x^2)^2, y(x),'implicit')

Maple raw output

y(x) = Int(-1/2/(x*(b-x)*(a-x))^(1/2)*(2*x^2+(-2*a-2*b)*x+a*b),x)+_C1, y(x) = In
t(1/2/(x*(b-x)*(a-x))^(1/2)*(2*x^2+(-2*a-2*b)*x+a*b),x)+_C1