Internal problem ID [3591]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 12
Problem number: 335.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
\[ \boxed {2 x \left (1-x \right ) y^{\prime }+\left (1-x \right ) y^{2}=-x} \]
✓ Solution by Maple
Time used: 0.046 (sec). Leaf size: 97
dsolve(2*x*(1-x)*diff(y(x),x)+x+(1-x)*y(x)^2 = 0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {x \left (\operatorname {LegendreQ}\left (-\frac {1}{2}, 1, \frac {-x +2}{x}\right ) c_{1} -\operatorname {LegendreQ}\left (\frac {1}{2}, 1, \frac {-x +2}{x}\right ) c_{1} +\operatorname {LegendreP}\left (-\frac {1}{2}, 1, \frac {-x +2}{x}\right )-\operatorname {LegendreP}\left (\frac {1}{2}, 1, \frac {-x +2}{x}\right )\right )}{2 \left (\operatorname {LegendreQ}\left (-\frac {1}{2}, 1, \frac {-x +2}{x}\right ) c_{1} +\operatorname {LegendreP}\left (-\frac {1}{2}, 1, \frac {-x +2}{x}\right )\right ) \left (x -1\right )} \]
✓ Solution by Mathematica
Time used: 0.806 (sec). Leaf size: 77
DSolve[2 x(1-x)y'[x]+x+(1-x)y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {2 \left (\pi G_{2,2}^{2,0}\left (x\left | \begin {array}{c} \frac {1}{2},\frac {3}{2} \\ 0,1 \\ \end {array} \right .\right )+c_1 (\operatorname {EllipticK}(x)-\operatorname {EllipticE}(x))\right )}{\pi G_{2,2}^{2,0}\left (x\left | \begin {array}{c} \frac {1}{2},\frac {3}{2} \\ 0,0 \\ \end {array} \right .\right )+2 c_1 \operatorname {EllipticE}(x)} y(x)\to 1-\frac {\operatorname {EllipticK}(x)}{\operatorname {EllipticE}(x)} \end{align*}