18.3.14 problem Problem 16.15

Internal problem ID [3514]
Book : Mathematical methods for physics and engineering, Riley, Hobson, Bence, second edition, 2002
Section : Chapter 16, Series solutions of ODEs. Section 16.6 Exercises, page 550
Problem number : Problem 16.15
Date solved : Sunday, March 30, 2025 at 01:45:29 AM
CAS classification : [_Gegenbauer, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 65

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

\[ y = \left (1-\frac {m^{2} z^{2}}{2}+\frac {m^{2} \left (m^{2}-4\right ) z^{4}}{24}\right ) y \left (0\right )+\left (z -\frac {\left (m^{2}-1\right ) z^{3}}{6}+\frac {\left (m^{4}-10 m^{2}+9\right ) z^{5}}{120}\right ) y^{\prime }\left (0\right )+O\left (z^{6}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 88

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

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

✓ Sympy. Time used: 0.952 (sec). Leaf size: 53

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

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