61.2.55 problem 55
Internal
problem
ID
[11982]
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
Date
solved
:
Friday, March 14, 2025 at 03:30:10 AM
CAS
classification
:
[_rational, _Riccati]
\begin{align*} \left (x^{2}-1\right ) y^{\prime }+\lambda \left (y^{2}-2 x y+1\right )&=0 \end{align*}
✓ Maple. Time used: 0.017 (sec). Leaf size: 280
ode:=(x^2-1)*diff(y(x),x)+lambda*(y(x)^2-2*x*y(x)+1) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = -\frac {2 \left (-\frac {x}{2}-\frac {1}{2}\right )^{2 \lambda } \left (\frac {x +1}{x -1}\right )^{\lambda } \left (\left (-\frac {x}{2}-\frac {1}{2}\right )^{-2 \lambda } \left (x +1\right ) \left (x -1\right )^{2} \operatorname {HeunCPrime}\left (0, 2 \lambda -1, 0, 0, \lambda ^{2}-\lambda +\frac {1}{2}, \frac {2}{x +1}\right )+8 \left (x -1\right )^{2} \operatorname {HeunCPrime}\left (0, -2 \lambda +1, 0, 0, \lambda ^{2}-\lambda +\frac {1}{2}, \frac {2}{x +1}\right ) c_{1} -8 \left (x +1\right )^{2} \left (\left (\frac {x +1}{x -1}\right )^{-\lambda } c_{1} \left (\left (\lambda -\frac {1}{2}\right ) x -\frac {\lambda }{2}+\frac {1}{2}\right ) \operatorname {hypergeom}\left (\left [1-\lambda , 1-\lambda \right ], \left [-2 \lambda +2\right ], -\frac {2}{x -1}\right )+\frac {\lambda \left (\frac {x +1}{x -1}\right )^{\lambda } \left (-\frac {x}{2}-\frac {1}{2}\right )^{-2 \lambda } \operatorname {hypergeom}\left (\left [\lambda , \lambda \right ], \left [2 \lambda \right ], -\frac {2}{x -1}\right ) \left (x -1\right )}{16}\right )\right )}{\lambda \left (x +1\right )^{2} \left (8 c_{1} \operatorname {hypergeom}\left (\left [1-\lambda , 1-\lambda \right ], \left [-2 \lambda +2\right ], -\frac {2}{x -1}\right ) \left (-\frac {x}{2}-\frac {1}{2}\right )^{2 \lambda }+\left (\frac {x +1}{x -1}\right )^{2 \lambda } \operatorname {hypergeom}\left (\left [\lambda , \lambda \right ], \left [2 \lambda \right ], -\frac {2}{x -1}\right ) \left (x -1\right )\right )}
\]
✓ Mathematica. Time used: 0.414 (sec). Leaf size: 47
ode=(x^2-1)*D[y[x],x]+\[Lambda]*(y[x]^2-2*x*y[x]+1)==0;
ic={};
DSolve[{ode,ic},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*}
✗ Sympy
from sympy import *
x = symbols("x")
cg = symbols("cg")
y = Function("y")
ode = Eq(cg*(-2*x*y(x) + y(x)**2 + 1) + (x**2 - 1)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out