53.1.218 problem 221

Internal problem ID [10690]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 221
Date solved : Tuesday, September 30, 2025 at 07:31:01 PM
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: 0.005 (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);
 

\[ y = 2 z^{2} \left ({\mathrm e}^{-2 z} c_2 \left (z -\frac {1}{2}\right ) \operatorname {Ei}_{1}\left (-2 z \right )+c_1 \left (z -\frac {1}{2}\right ) {\mathrm e}^{-2 z}+\frac {c_2}{2}\right ) \]

✓ Mathematica. Time used: 0.556 (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]
 

\begin{align*} 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 ) \end{align*}

✓ Sympy. Time used: 0.902 (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)
 

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