87.24.10 problem 10

Internal problem ID [23792]
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 : 10
Date solved : Thursday, October 02, 2025 at 09:45:14 PM
CAS classification : [_Bessel]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 73

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

\[ y = c_1 \,x^{-p} \left (1+\frac {1}{4 p -4} x^{2}+\frac {1}{32} \frac {1}{\left (p -2\right ) \left (p -1\right )} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \,x^{p} \left (1-\frac {1}{4 p +4} x^{2}+\frac {1}{32} \frac {1}{\left (p +2\right ) \left (p +1\right )} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 95

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

\[ y(x)\to c_2 \left (\frac {p^4 x^5}{120}+\frac {p^2 x^5}{30}+\frac {p^2 x^3}{6}-\frac {x^5}{40}+\frac {x^3}{6}+x\right )+c_1 \left (\frac {p^4 x^4}{24}+\frac {p^2 x^4}{12}+\frac {p^2 x^2}{2}-\frac {x^4}{12}+1\right ) \]

✓ Sympy. Time used: 0.340 (sec). Leaf size: 71

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

\[ y{\left (x \right )} = - \frac {3 x^{5} r{\left (3 \right )}}{20} + \frac {p^{2} x^{5} r{\left (3 \right )}}{20} + C_{2} \left (\frac {p^{4} x^{4}}{24} - \frac {p^{2} x^{4}}{12} + \frac {p^{2} x^{2}}{2} - \frac {x^{4}}{12} + 1\right ) + C_{1} x \left (1 - \frac {x^{4}}{20}\right ) + O\left (x^{6}\right ) \]