59.1.634 problem 651

Internal problem ID [9806]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 651
Date solved : Sunday, March 30, 2025 at 02:47:02 PM
CAS classification : [_Gegenbauer]

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

✓ Maple. Time used: 0.038 (sec). Leaf size: 45

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

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

✓ Mathematica. Time used: 0.061 (sec). Leaf size: 90

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

\[ y(z)\to \frac {\sqrt {2} c_1 \cos \left (2 \sqrt {2} \arcsin \left (\frac {\sqrt {1-z}}{\sqrt {2}}\right )\right )+\sqrt {\pi } c_2 \sqrt [4]{1-z^2} Q_{-\frac {1}{2}+\sqrt {2}}^{\frac {1}{2}}(z)}{\sqrt {\pi } \sqrt [4]{-\left (z^2-1\right )^2}} \]

✓ Sympy. Time used: 0.868 (sec). Leaf size: 31

from sympy import * 
z = symbols("z") 
y = Function("y") 
ode = Eq(-3*z*Derivative(y(z), z) + (1 - z**2)*Derivative(y(z), (z, 2)) + y(z),0) 
ics = {} 
dsolve(ode,func=y(z),ics=ics)
 

\[ y{\left (z \right )} = C_{2} \left (- \frac {7 z^{4}}{24} - \frac {z^{2}}{2} + 1\right ) + C_{1} z \left (\frac {z^{2}}{3} + 1\right ) + O\left (z^{6}\right ) \]