18.3.1 problem Problem 16.1

Internal problem ID [3501]
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 16.1
Date solved : Tuesday, March 04, 2025 at 04:43:41 PM
CAS classification : [_Gegenbauer]

\begin{align*} \left (-z^{2}+1\right ) y^{\prime \prime }-3 z y^{\prime }+\lambda y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 53

Order:=6; 
ode:=(-z^2+1)*diff(diff(y(z),z),z)-3*z*diff(y(z),z)+lambda*y(z) = 0; 
dsolve(ode,y(z),type='series',z=0);
 

\[ y \left (z \right ) = \left (1-\frac {\lambda \,z^{2}}{2}+\frac {\lambda \left (\lambda -8\right ) z^{4}}{24}\right ) y \left (0\right )+\left (z -\frac {\left (\lambda -3\right ) z^{3}}{6}+\frac {\left (\lambda -3\right ) \left (\lambda -15\right ) z^{5}}{120}\right ) D\left (y \right )\left (0\right )+O\left (z^{6}\right ) \]

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

ode=(1-z^2)*D[y[z],{z,2}]-3*z*D[y[z],z]+\[Lambda]*y[z]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[z],{z,0,5}]
 

\[ y(z)\to c_2 \left (\frac {\lambda ^2 z^5}{120}-\frac {3 \lambda z^5}{20}+\frac {3 z^5}{8}-\frac {\lambda z^3}{6}+\frac {z^3}{2}+z\right )+c_1 \left (\frac {\lambda ^2 z^4}{24}-\frac {\lambda z^4}{3}-\frac {\lambda z^2}{2}+1\right ) \]

✓ Sympy. Time used: 0.913 (sec). Leaf size: 48

from sympy import * 
z = symbols("z") 
cg = symbols("cg") 
y = Function("y") 
ode = Eq(cg*y(z) - 3*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)
 

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