82.3.28 problem 25-28

Internal problem ID [21811]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 25. Power series about a singular point. Page 762
Problem number : 25-28
Date solved : Thursday, October 02, 2025 at 08:02:30 PM
CAS classification : [_Gegenbauer]

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

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 51

Order:=6; 
ode:=(-x^2+1)*diff(diff(y(x),x),x)+2*x*diff(y(x),x)-lambda*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1+\frac {\lambda \,x^{2}}{2}+\frac {\lambda \left (\lambda -2\right ) x^{4}}{24}\right ) y \left (0\right )+\left (x +\frac {\left (\lambda -2\right ) x^{3}}{6}+\frac {\lambda \left (\lambda -2\right ) x^{5}}{120}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 73

ode=(1-x^2)*D[y[x],{x,2}]+2*x*D[y[x],x]- \[Lambda]*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {\lambda ^2 x^5}{120}-\frac {\lambda x^5}{60}+\frac {\lambda x^3}{6}-\frac {x^3}{3}+x\right )+c_1 \left (\frac {\lambda ^2 x^4}{24}-\frac {\lambda x^4}{12}+\frac {\lambda x^2}{2}+1\right ) \]

✓ Sympy. Time used: 0.333 (sec). Leaf size: 48

from sympy import * 
x = symbols("x") 
lambda_ = symbols("lambda_") 
y = Function("y") 
ode = Eq(-lambda_*y(x) + 2*x*Derivative(y(x), x) + (1 - x**2)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {\lambda _{}^{2} x^{4}}{24} - \frac {\lambda _{} x^{4}}{12} + \frac {\lambda _{} x^{2}}{2} + 1\right ) + C_{1} x \left (\frac {\lambda _{} x^{2}}{6} - \frac {x^{2}}{3} + 1\right ) + O\left (x^{6}\right ) \]