29.12.16 problem 335

Internal problem ID [4935]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 12
Problem number : 335
Date solved : Monday, January 27, 2025 at 09:52:40 AM
CAS classification : [_rational, _Riccati]

\begin{align*} 2 x \left (1-x \right ) y^{\prime }+x +\left (1-x \right ) y^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.146 (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: 16.863 (sec). Leaf size: 77

DSolve[2 x(1-x)D[y[x],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*}