59.1.214 problem 217

Internal problem ID [9386]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 217
Date solved : Sunday, March 30, 2025 at 02:33:45 PM
CAS classification : [_Gegenbauer]

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

✓ Maple. Time used: 0.049 (sec). Leaf size: 49

ode:=(-z^2+1)*diff(diff(y(z),z),z)-3*z*diff(y(z),z)+lambda*y(z) = 0; 
dsolve(ode,y(z), singsol=all);
 

\[ y = \frac {c_2 \left (z +\sqrt {z^{2}-1}\right )^{-\sqrt {\lambda +1}}+c_1 \left (z +\sqrt {z^{2}-1}\right )^{\sqrt {\lambda +1}}}{\sqrt {z^{2}-1}} \]

✓ Mathematica. Time used: 0.04 (sec). Leaf size: 54

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

\[ y(z)\to \frac {c_1 P_{\sqrt {\lambda +1}-\frac {1}{2}}^{\frac {1}{2}}(z)+c_2 Q_{\sqrt {\lambda +1}-\frac {1}{2}}^{\frac {1}{2}}(z)}{\sqrt [4]{z^2-1}} \]

✗ Sympy

from sympy import * 
z = symbols("z") 
lambda_ = symbols("lambda_") 
y = Function("y") 
ode = Eq(lambda_*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)
 

False