73.23.26 problem 33.10

Internal problem ID [15601]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 33. Power series solutions I: Basic computational methods. Additional Exercises. page 641
Problem number : 33.10
Date solved : Monday, March 31, 2025 at 01:42:07 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: 0.007 (sec). Leaf size: 53

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);
 

\[ y = \left (1-\frac {\lambda \,x^{2}}{2}+\frac {\lambda \left (\lambda -6\right ) x^{4}}{24}\right ) y \left (0\right )+\left (x -\frac {\left (\lambda -2\right ) x^{3}}{6}+\frac {\left (\lambda -2\right ) \left (\lambda -12\right ) x^{5}}{120}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 80

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}]
 

\[ y(x)\to c_2 \left (\frac {\lambda ^2 x^5}{120}-\frac {7 \lambda x^5}{60}+\frac {x^5}{5}-\frac {\lambda x^3}{6}+\frac {x^3}{3}+x\right )+c_1 \left (\frac {\lambda ^2 x^4}{24}-\frac {\lambda x^4}{4}-\frac {\lambda x^2}{2}+1\right ) \]

✓ Sympy. Time used: 0.880 (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)
 

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