79.16.1 problem (a)

Internal problem ID [21118]
Book : Ordinary Differential Equations. By Wolfgang Walter. Graduate texts in Mathematics. Springer. NY. QA372.W224 1998
Section : Chapter V. Complex Linear Systems. Excercise XII at page 244
Problem number : (a)
Date solved : Thursday, October 02, 2025 at 07:08:36 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 21

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

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

✓ Mathematica. Time used: 0.051 (sec). Leaf size: 27

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

\begin{align*} u(z)&\to \frac {e^{\frac {1}{z}} \left (c_1-c_2 \operatorname {ExpIntegralEi}\left (-\frac {1}{z}\right )\right )}{z} \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(u(z), z) - (-z**2*Derivative(u(z), (z, 2)) - u(z))/(3*z + 1) cannot be solved by the factorable group method