61.32.11 problem 221

Internal problem ID [12642]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-7
Problem number : 221
Date solved : Wednesday, March 05, 2025 at 08:14:41 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 55

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

\[ y = \sqrt {x^{2}-1}\, \left (\left (\frac {x -1}{x +1}\right )^{-\frac {\sqrt {-a +1}}{2}} c_{2} +\left (\frac {x -1}{x +1}\right )^{\frac {\sqrt {-a +1}}{2}} c_{1} \right ) \]

✓ Mathematica. Time used: 0.073 (sec). Leaf size: 82

ode=(x^2-1)^2*D[y[x],{x,2}]+a*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \exp \left (\int _1^x\frac {K[1]+\sqrt {1-a}}{K[1]^2-1}dK[1]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[2]}\frac {K[1]+\sqrt {1-a}}{K[1]^2-1}dK[1]\right )dK[2]+c_1\right ) \]

✓ Sympy. Time used: 0.371 (sec). Leaf size: 100

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a*y(x) + (x**2 - 1)**2*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {\sqrt {x^{2} - 1} \left (C_{1} \sqrt {x^{2}} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2} - \frac {\sqrt {1 - a}}{2}, 1 - \frac {\sqrt {1 - a}}{2} \\ \frac {3}{2} \end {matrix}\middle | {x^{2}} \right )} + C_{2} {{}_{2}F_{1}\left (\begin {matrix} - \frac {\sqrt {1 - a}}{2}, \frac {1}{2} - \frac {\sqrt {1 - a}}{2} \\ \frac {1}{2} \end {matrix}\middle | {x^{2}} \right )}\right ) \sqrt [4]{x^{2}} e^{- \frac {\sqrt {1 - a} \log {\left (x^{2} - 1 \right )}}{2}}}{\sqrt {x}} \]