60.1.165 problem 166

Internal problem ID [10179]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 166
Date solved : Monday, January 27, 2025 at 06:31:29 PM
CAS classification : [_rational, _Riccati]

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

✓ Solution by Maple

Time used: 0.087 (sec). Leaf size: 97

dsolve(2*x*(x-1)*diff(y(x),x) + (x-1)*y(x)^2 - x=0,y(x), singsol=all)
 

\[ y = \frac {x \left (\operatorname {LegendreQ}\left (-\frac {1}{2}, 1, \frac {-x +2}{x}\right ) c_{1} -\operatorname {LegendreQ}\left (\frac {1}{2}, 1, \frac {-x +2}{x}\right ) c_{1} +\operatorname {LegendreP}\left (-\frac {1}{2}, 1, \frac {-x +2}{x}\right )-\operatorname {LegendreP}\left (\frac {1}{2}, 1, \frac {-x +2}{x}\right )\right )}{2 \left (\operatorname {LegendreQ}\left (-\frac {1}{2}, 1, \frac {-x +2}{x}\right ) c_{1} +\operatorname {LegendreP}\left (-\frac {1}{2}, 1, \frac {-x +2}{x}\right )\right ) \left (x -1\right )} \]

✓ Solution by Mathematica

Time used: 16.673 (sec). Leaf size: 77

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