82.2.16 problem 24-15

Internal problem ID [21777]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 24. Power series about an ordinary point. Page 719
Problem number : 24-15
Date solved : Thursday, October 02, 2025 at 08:02:02 PM
CAS classification : [_Gegenbauer]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 81

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

\[ y = \left (1-\frac {\lambda \left (\lambda +1\right ) x^{2}}{2}+\frac {\lambda \left (\lambda ^{3}+2 \lambda ^{2}-5 \lambda -6\right ) x^{4}}{24}\right ) y \left (0\right )+\left (x -\frac {\left (\lambda ^{2}+\lambda -2\right ) x^{3}}{6}+\frac {\left (\lambda ^{4}+2 \lambda ^{3}-13 \lambda ^{2}-14 \lambda +24\right ) x^{5}}{120}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 127

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

\[ y(x)\to c_2 \left (\frac {1}{60} \left (-\lambda ^2-\lambda \right ) x^5-\frac {1}{120} \left (-\lambda ^2-\lambda \right ) \left (\lambda ^2+\lambda \right ) x^5-\frac {1}{10} \left (\lambda ^2+\lambda \right ) x^5+\frac {x^5}{5}-\frac {1}{6} \left (\lambda ^2+\lambda \right ) x^3+\frac {x^3}{3}+x\right )+c_1 \left (\frac {1}{24} \left (\lambda ^2+\lambda \right )^2 x^4-\frac {1}{4} \left (\lambda ^2+\lambda \right ) x^4-\frac {1}{2} \left (\lambda ^2+\lambda \right ) x^2+1\right ) \]

✓ Sympy. Time used: 0.451 (sec). Leaf size: 83

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(lambda_*(lambda_ + 1)*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)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {\lambda _{}^{4} x^{4}}{24} + \frac {\lambda _{}^{3} x^{4}}{12} - \frac {5 \lambda _{}^{2} x^{4}}{24} - \frac {\lambda _{}^{2} x^{2}}{2} - \frac {\lambda _{} x^{4}}{4} - \frac {\lambda _{} x^{2}}{2} + 1\right ) + C_{1} x \left (- \frac {\lambda _{}^{2} x^{2}}{6} - \frac {\lambda _{} x^{2}}{6} + \frac {x^{2}}{3} + 1\right ) + O\left (x^{6}\right ) \]