59.1.294 problem 297
Internal
problem
ID
[9466]
Book
:
Collection
of
Kovacic
problems
Section
:
section
1
Problem
number
:
297
Date
solved
:
Sunday, March 30, 2025 at 02:35:33 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} \left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime }+y&=0 \end{align*}
✓ Maple. Time used: 0.024 (sec). Leaf size: 66
ode:=(-x^2+1)*diff(diff(y(x),x),x)-diff(y(x),x)+y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = c_1 \operatorname {hypergeom}\left (\left [-\frac {1}{2}-\frac {\sqrt {5}}{2}, \frac {\sqrt {5}}{2}-\frac {1}{2}\right ], \left [-\frac {1}{2}\right ], \frac {1}{2}+\frac {x}{2}\right )+2 c_2 \sqrt {2+2 x}\, \operatorname {hypergeom}\left (\left [1-\frac {\sqrt {5}}{2}, \frac {\sqrt {5}}{2}+1\right ], \left [\frac {5}{2}\right ], \frac {1}{2}+\frac {x}{2}\right ) \left (x +1\right )
\]
✓ Mathematica. Time used: 6.1 (sec). Leaf size: 210
ode=(1-x^2)*D[y[x],{x,2}]-D[y[x],x]+y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \left (\sqrt {x-1}-\sqrt {x+1}\right )^{-\frac {1}{2}-\frac {\sqrt {5}}{2}} \left (\sqrt {x-1}+\sqrt {x+1}\right )^{\frac {1}{2} \left (\sqrt {5}-1\right )} \left (\sqrt {x-1}-\sqrt {5} \sqrt {x+1}\right ) \left (c_2 \int _1^x-\frac {2 e^{\text {arctanh}(K[2])} \left (\sqrt {K[2]-1}-\sqrt {K[2]+1}\right )^{\sqrt {5}} \left (\sqrt {K[2]-1}+\sqrt {K[2]+1}\right )^{-\sqrt {5}}}{\left (\sqrt {K[2]-1}-\sqrt {5} \sqrt {K[2]+1}\right )^2}dK[2]+c_1\right ) \exp \left (-\frac {1}{2} \int _1^x\frac {1}{K[1]^2-1}dK[1]-\frac {\text {arctanh}(x)}{2}\right )
\]
✓ Sympy. Time used: 0.612 (sec). Leaf size: 122
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((1 - x**2)*Derivative(y(x), (x, 2)) + y(x) - Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \frac {\sqrt [4]{x - 1} \sqrt [4]{x^{2} - 1} \left (x^{2} - 1\right )^{\frac {\sqrt {5}}{4}} \left (C_{1} \left (\frac {x + 1}{x - 1}\right )^{\frac {3}{2}} {{}_{2}F_{1}\left (\begin {matrix} 1 - \frac {\sqrt {5}}{2}, \frac {3}{2} - \frac {\sqrt {5}}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {x + 1}{x - 1}} \right )} + C_{2} {{}_{2}F_{1}\left (\begin {matrix} - \frac {\sqrt {5}}{2}, - \frac {\sqrt {5}}{2} - \frac {1}{2} \\ - \frac {1}{2} \end {matrix}\middle | {\frac {x + 1}{x - 1}} \right )}\right ) e^{\frac {\sqrt {5} \left (\log {\left (x - 1 \right )} - \log {\left (x + 1 \right )}\right )}{4}}}{\sqrt [4]{x + 1}}
\]