29.11.7 problem 298
Internal
problem
ID
[4898]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
11
Problem
number
:
298
Date
solved
:
Sunday, March 30, 2025 at 04:09:34 AM
CAS
classification
:
[_rational, _Riccati]
\begin{align*} \left (-x^{2}+1\right ) y^{\prime }&=n \left (1-2 x y+y^{2}\right ) \end{align*}
✓ Maple. Time used: 0.026 (sec). Leaf size: 280
ode:=(-x^2+1)*diff(y(x),x) = n*(1-2*x*y(x)+y(x)^2);
dsolve(ode,y(x), singsol=all);
\[
y = \frac {2 \left (-\frac {\left (-\frac {x}{2}-\frac {1}{2}\right )^{-2 n} \left (x +1\right ) \left (x -1\right )^{2} \operatorname {HeunCPrime}\left (0, 2 n -1, 0, 0, n^{2}-n +\frac {1}{2}, \frac {2}{x +1}\right )}{8}-\left (x -1\right )^{2} \operatorname {HeunCPrime}\left (0, -2 n +1, 0, 0, n^{2}-n +\frac {1}{2}, \frac {2}{x +1}\right ) c_1 +\left (x +1\right )^{2} \left (c_1 \left (\left (n -\frac {1}{2}\right ) x -\frac {n}{2}+\frac {1}{2}\right ) \left (\frac {x +1}{x -1}\right )^{-n} \operatorname {hypergeom}\left (\left [-n +1, -n +1\right ], \left [-2 n +2\right ], -\frac {2}{x -1}\right )+\frac {n \left (-\frac {x}{2}-\frac {1}{2}\right )^{-2 n} \operatorname {hypergeom}\left (\left [n , n\right ], \left [2 n \right ], -\frac {2}{x -1}\right ) \left (\frac {x +1}{x -1}\right )^{n} \left (x -1\right )}{16}\right )\right ) \left (\frac {x +1}{x -1}\right )^{n} \left (-\frac {x}{2}-\frac {1}{2}\right )^{2 n}}{\left (x +1\right )^{2} n \left (\operatorname {hypergeom}\left (\left [-n +1, -n +1\right ], \left [-2 n +2\right ], -\frac {2}{x -1}\right ) c_1 \left (-\frac {x}{2}-\frac {1}{2}\right )^{2 n}+\frac {\operatorname {hypergeom}\left (\left [n , n\right ], \left [2 n \right ], -\frac {2}{x -1}\right ) \left (\frac {x +1}{x -1}\right )^{2 n} \left (x -1\right )}{8}\right )}
\]
✓ Mathematica. Time used: 0.342 (sec). Leaf size: 47
ode=(1-x^2)*D[y[x],x]==n*(1-2*x*y[x]+y[x]^2);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {\operatorname {LegendreQ}(n,x)+c_1 \operatorname {LegendreP}(n,x)}{\operatorname {LegendreQ}(n-1,x)+c_1 \operatorname {LegendreP}(n-1,x)} \\
y(x)\to \frac {\operatorname {LegendreP}(n,x)}{\operatorname {LegendreP}(n-1,x)} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
n = symbols("n")
y = Function("y")
ode = Eq(-n*(-2*x*y(x) + y(x)**2 + 1) + (1 - x**2)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out