Internal problem ID [10385]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.2. Riccati Equation. 1.2.2. Equations Containing Power
Functions
Problem number: 55.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
\[ \boxed {\left (x^{2}-1\right ) y^{\prime }+\lambda \left (y^{2}-2 x y+1\right )=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 280
dsolve((x^2-1)*diff(y(x),x)+lambda*(y(x)^2-2*x*y(x)+1)=0,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {2 \left (\left (-\frac {x}{2}-\frac {1}{2}\right )^{-2 \lambda } \left (1+x \right ) \left (-1+x \right )^{2} \operatorname {HeunCPrime}\left (0, 2 \lambda -1, 0, 0, \lambda ^{2}-\lambda +\frac {1}{2}, \frac {2}{1+x}\right )+8 \left (-1+x \right )^{2} \operatorname {HeunCPrime}\left (0, -2 \lambda +1, 0, 0, \lambda ^{2}-\lambda +\frac {1}{2}, \frac {2}{1+x}\right ) c_{1} -8 \left (1+x \right )^{2} \left (\left (\left (\lambda -\frac {1}{2}\right ) x -\frac {\lambda }{2}+\frac {1}{2}\right ) c_{1} \left (\frac {1+x}{-1+x}\right )^{-\lambda } \operatorname {hypergeom}\left (\left [1-\lambda , 1-\lambda \right ], \left [-2 \lambda +2\right ], -\frac {2}{-1+x}\right )+\frac {\lambda \left (\frac {1+x}{-1+x}\right )^{\lambda } \left (-\frac {x}{2}-\frac {1}{2}\right )^{-2 \lambda } \operatorname {hypergeom}\left (\left [\lambda , \lambda \right ], \left [2 \lambda \right ], -\frac {2}{-1+x}\right ) \left (-1+x \right )}{16}\right )\right ) \left (-\frac {x}{2}-\frac {1}{2}\right )^{2 \lambda } \left (\frac {1+x}{-1+x}\right )^{\lambda }}{\left (8 c_{1} \operatorname {hypergeom}\left (\left [1-\lambda , 1-\lambda \right ], \left [-2 \lambda +2\right ], -\frac {2}{-1+x}\right ) \left (-\frac {x}{2}-\frac {1}{2}\right )^{2 \lambda }+\operatorname {hypergeom}\left (\left [\lambda , \lambda \right ], \left [2 \lambda \right ], -\frac {2}{-1+x}\right ) \left (\frac {1+x}{-1+x}\right )^{2 \lambda } \left (-1+x \right )\right ) \left (1+x \right )^{2} \lambda } \]
✓ Solution by Mathematica
Time used: 0.643 (sec). Leaf size: 47
DSolve[(x^2-1)*y'[x]+\[Lambda]*(y[x]^2-2*x*y[x]+1)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\operatorname {LegendreQ}(\lambda ,x)+c_1 \operatorname {LegendreP}(\lambda ,x)}{\operatorname {LegendreQ}(\lambda -1,x)+c_1 \operatorname {LegendreP}(\lambda -1,x)} \\ y(x)\to \frac {\operatorname {LegendreP}(\lambda ,x)}{\operatorname {LegendreP}(\lambda -1,x)} \\ \end{align*}