Internal problem ID [3658]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 31
Problem number: 932.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_quadrature]
Solve \begin {gather*} \boxed {4 x \left (-x +a \right ) \left (-x +b \right ) \left (y^{\prime }\right )^{2}-\left (a b -2 \left (a +b \right ) x +2 x^{2}\right )^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.107 (sec). Leaf size: 81
dsolve(4*x*(a-x)*(b-x)*diff(y(x),x)^2 = (a*b-2*x*(a+b)+2*x^2)^2,y(x), singsol=all)
\begin{align*} y \relax (x ) = \int -\frac {a b -2 a x -2 b x +2 x^{2}}{2 \sqrt {x \left (-x +b \right ) \left (-x +a \right )}}d x +c_{1} \\ y \relax (x ) = \int \frac {a b -2 a x -2 b x +2 x^{2}}{2 \sqrt {x \left (-x +b \right ) \left (-x +a \right )}}d x +c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 20.117 (sec). Leaf size: 299
DSolve[4 x(a-x)(b-x) (y'[x])^2==(a b-2 x(a+b)+2 x^2)^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x \sqrt {1-\frac {a}{x}} \sqrt {\frac {x-b}{a-b}} \left (b (a+2 b) \text {EllipticF}\left (i \sinh ^{-1}\left (\sqrt {\frac {x}{a}-1}\right ),\frac {a}{a-b}\right )+2 (a-b) (a+b) E\left (i \sinh ^{-1}\left (\sqrt {\frac {x}{a}-1}\right )|\frac {a}{a-b}\right )\right )}{3 \sqrt {x (a-x) (x-b)}}+\frac {2}{3} i \sqrt {x (a-x) (x-b)}+c_1 \\ y(x)\to -\frac {x \sqrt {1-\frac {a}{x}} \sqrt {\frac {x-b}{a-b}} \left (b (a+2 b) \text {EllipticF}\left (i \sinh ^{-1}\left (\sqrt {\frac {x}{a}-1}\right ),\frac {a}{a-b}\right )+2 (a-b) (a+b) E\left (i \sinh ^{-1}\left (\sqrt {\frac {x}{a}-1}\right )|\frac {a}{a-b}\right )\right )}{3 \sqrt {x (a-x) (x-b)}}-\frac {2}{3} i \sqrt {x (a-x) (x-b)}+c_1 \\ \end{align*}