59.1.218 problem 221

Internal problem ID [9390]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 221
Date solved : Wednesday, March 05, 2025 at 07:48:30 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} z y^{\prime \prime }+\left (2 z -3\right ) y^{\prime }+\frac {4 y}{z}&=0 \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 36

ode:=z*diff(diff(y(z),z),z)+(2*z-3)*diff(y(z),z)+4/z*y(z) = 0; 
dsolve(ode,y(z), singsol=all);
 

\[ y \left (z \right ) = 2 \left (c_{2} {\mathrm e}^{-2 z} \left (-\frac {1}{2}+z \right ) \operatorname {Ei}_{1}\left (-2 z \right )+c_{1} \left (-\frac {1}{2}+z \right ) {\mathrm e}^{-2 z}+\frac {c_{2}}{2}\right ) z^{2} \]

✓ Mathematica. Time used: 0.939 (sec). Leaf size: 55

ode=z*D[y[z],{z,2}]+(2*z-3)*D[y[z],z]+4/z*y[z]==0; 
ic={}; 
DSolve[{ode,ic},y[z],z,IncludeSingularSolutions->True]
 

\[ y(z)\to \frac {1}{2} e^{-2 z} z^2 (2 z-1) \left (c_2 \int _1^z\frac {4 e^{2 K[1]}}{(1-2 K[1])^2 K[1]}dK[1]+c_1\right ) \]

✓ Sympy. Time used: 1.721 (sec). Leaf size: 416

from sympy import * 
z = symbols("z") 
y = Function("y") 
ode = Eq(z*Derivative(y(z), (z, 2)) + (2*z - 3)*Derivative(y(z), z) + 4*y(z)/z,0) 
ics = {} 
dsolve(ode,func=y(z),ics=ics)
 

\[ \text {Solution too large to show} \]