87.24.11 problem 11

Internal problem ID [23793]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 5. Series solutions of second order linear equations. Exercise at page 218
Problem number : 11
Date solved : Thursday, October 02, 2025 at 09:45:15 PM
CAS classification : [_Gegenbauer]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 60

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

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

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 78

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

\[ y(x)\to c_2 \left (\frac {p^4 x^5}{120}-\frac {p^2 x^5}{60}-\frac {p^2 x^3}{6}-\frac {x^5}{40}-\frac {x^3}{6}+x\right )+c_1 \left (\frac {p^4 x^4}{24}-\frac {p^2 x^2}{2}+1\right ) \]

✓ Sympy. Time used: 0.332 (sec). Leaf size: 44

from sympy import * 
x = symbols("x") 
p = symbols("p") 
y = Function("y") 
ode = Eq(p**2*y(x) + 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)
 

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