23.3.474 problem 480

Internal problem ID [6188]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 480
Date solved : Tuesday, September 30, 2025 at 02:35:59 PM
CAS classification : [_Gegenbauer]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 31

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

\[ y = -\frac {c_2 a \ln \left (a x +1\right ) x}{2}+\frac {c_2 a \ln \left (a x -1\right ) x}{2}+c_1 x +c_2 \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 39

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

\begin{align*} y(x)&\to a c_1 x-\frac {1}{2} c_2 (a x \log (1-a x)-a x \log (a x+1)+2) \end{align*}

✓ Sympy. Time used: 0.363 (sec). Leaf size: 104

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

\[ y{\left (x \right )} = \frac {\left (a^{2} x^{2} - 1\right )^{\frac {3}{4}} \left (C_{1} \sqrt {\frac {a^{2} x^{2}}{a^{2} x^{2} - 1}} {{}_{2}F_{1}\left (\begin {matrix} 0, 0 \\ \frac {3}{2} \end {matrix}\middle | {\frac {a^{2} x^{2}}{a^{2} x^{2} - 1}} \right )} + C_{2} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{2} \\ \frac {1}{2} \end {matrix}\middle | {\frac {a^{2} x^{2}}{a^{2} x^{2} - 1}} \right )}\right ) \sqrt [4]{x^{2}}}{\sqrt {x} \sqrt [4]{x^{2} - \frac {1}{a^{2}}}} \]